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Physics 201, Principles of Physics I

Assignments Table, Full Version

It is assumed that you have either completed the tasks specified in the Initial Activities (these are specified, in large red letters, in the menu in the left-hand frame of your Homepage), or are in the process of reviewing this document according to instructions in those activities.

You will not understand how to complete or submit assignments until you previously completed the tasks in the Initial Activities.

Table of Assignments, Topics and Specific Objectives

How to Complete Assignments
Module 1: Assignments 0-8
Module 2: Assignments 9-20
Module 3: Assignments 21-32
Module 4: Assignments 33-40
Brief Assignments Page

Text and labs are bottom line.

1st phase: learn about motion and mathematical techniques, experience basic systems, get up to speed while getting used to the text (assts 1-6)

Videos are embedded in the document, which contains a series of class notes from an earlier class. You can access these notes at the link

Class Notes

This link also appears as the heading in the 8th column of the Assignments Page. The numbers in that column refer to the number of the specific Class Notes document corresponding to an assignment.

Some assignments are primarily associated with more than one Class Notes document, while some assignments do not mention a specific Class Notes document.

How to Complete Assignments

Each assignment is preceded by the following row of headings:

Asst

seed

Introductory Problem Sets

lab/activity

text

outline

Class Notes

other

query

The same headings are expanded below, with some explanation:

Asst

(the number of the asst)

(qa document to worked through and submitted at the beginning of the asst)

seed

(questions to be opened, worked and submitted)

Introductory Problem Sets

(worked problems with explanations, to be completely mastered as a rudimentary core of understanding)

lab/activity

(instructions for hands-on activities and labs, to be opened, conducted using lab materials, and submitted)

text

(assignment from the text, including assigned text sections and end-of-chapter problems)

outline

(there might not be any information under this heading; outline is now provided for each module)

Class Notes

(click on the link and view the Class Notes (video links do not work online), or run from DVD for version with working video links; take notes)

other

(miscellaneous information to help organize your knowledge, as well as some problems to be submitted early in the course)

query

(document to be submitted at end of assignment, will ask about various things done in assignment, including selected assigned problems, labs, activities, class notes)

Order of tasks:

As should be clear from the above, you start by submitting the q_a_ document and end by submitting the Query document.
It is recommended that you do the qa, then the 'seed' questions, then Introductory Problem Sets first.
The remaining tasks may be done in any order.

Objectives:

Specific objectives for each assignment are listed in the row below the assignment. They are listed after the assignment because only in working through the assignment can you be expected to understand all the terminology used in stating the objectives. The objectives are intended to focus what you experience.
Learning not a linear process. Most of the objectives in this course take time to 'sink in'. Full understanding of an objective first stated in a given assignment gradually develops, beginning in that assignment and continuing as it is enriched and reinforced through at least a few subsequent assignments.

Table of Assignments and Specific Objectives

This is the official assignments table for the course. This version contains a great deal of information about goals, specific objectives and contents of various documents. There is, in fact, more information than you will need for routine use.

The Brief Assignments Page (see the menu in the Contents frame) might be more convenient for day-to-day use.

Symbols used in this course:

Note that symbols might not be correctly represented by your browser. For this reason the Greek letters will be spelled out, with ` in front of the spelling. You should substitute the appropriate symbol when making notes.

The table below might or might not represent the symbols correctly:

`alpha a

`beta b

`d or `Delta D

`delta d

`epsilon e

`lambda l

`omega w

`rho r

`theta q

`mu m

`pi p

`phi f

`psi y

`gamma g

`sigma s

`Sigma S

`tau t

Module 1: Preliminary Assignment - Assignment 8

Major Quiz over Module 1 is assigned as part of Assignment 10

Summary of Main Ideas in Module 1

Having completed and reviewed your work in the assignments comprising this Module you should understand these ideas and be able to apply them to solving problems and analyzing real-world situations.

Idea 1: Units and Rates are Keys to Understanding

When we do calculations we always write down and think about the meaning of the units of the quantities involved. This is a very time-effective strategy for catching errors and for developing the ability to think through and solve problems.

We also do the algebra involved in determining the units of our results.

Simple rates are easy and natural if they concern money and the rate at which it is earned, and fairly easy in the context of velocity as rate of change of position. While thinking in the context of money or motion, simple rate problems can be solved using common sense.

Definition of an average rate: The average rate of change of A with respect to B is

average rate = change in A / change in B

** Rate problems develop into the idea of the derivative in calculus, and into differential equations. These concepts are much easier if we think in the context of simple rates.

An average rate of change of a quantity with respect to clock time is represented as the slope between two points of a graph of that quantity vs. clock time.

The change of a quantity during a short time interval is approximated by the area under the trapezoid formed by the graph of the rate of change of the quantity with respect to clock time vs. time clock time, for that time interval.

Idea 2: Velocity and Acceleration as Rates

Velocity and acceleration are defined as rates:

The average velocity of an object between two clock times is the average rate at which its position changes between those clock times. The average rate at which position changes is ave rate = change in position / change in clock time = ave velocity.

The average velocity of an object moving along a straight line is therefore represented by the slope of a graph of position vs. clock time.

** The velocity function v(t) is the derivative of the position function s(t) for and object.

The average acceleration of an object between two clock times is the average rate at which its velocity changes between those clock times. The average rate at which velocity changes is ave rate = change in velocity / change in clock time = ave acceleration.

The average acceleration of an object moving along a straight line is therefore represented by the slope of a graph of velocity vs. clock time.

** The acceleration function a(t) is the derivative of the velocity function v(t) for and object.

Idea 3: Applying the Ideas of Rates to Position, Velocity and Acceleration

All the concepts of rates apply to position, velocity and acceleration and their relationships.

Applying the concepts of rates we see that changes in position and velocity can be obtained from knowledge of velocities and accelerations, respectively:

The change in the position of an object over a given time interval is equal to the product of its average velocity and the duration of the time interval.

The change in position of an object moving along a straight line is therefore approximated as the area under the appropriate trapezoid on a graph of velocity vs. clock time.

The precise change in position is equal to the exact area under the graph of velocity vs. clock time, between the two given clock times.

** The change in position `ds between two clock times is the integral of the velocity function v(t) between the two corresponding clock times.

The change in the velocity of an object over a given time interval is equal to the product of its average acceleration and the duration of the time interval.

The change in velocity of an object moving along a straight line is therefore approximated as the area under the appropriate trapezoid on a graph of acceleration vs. clock time.

The precise change in velocity is equal to the exact area under the graph of acceleration vs. clock time, between the two given clock times.

** The change in velocity `dv between two clock times is the integral of the acceleration function a(t) between the two corresponding clock times.

The velocity of an object may change at a constant, an increasing, or a decreasing rate; the graph of velocity vs. clock time will reveal which.

Idea 4: Uniform Acceleration

If the velocity of an object changes at a constant rate, i.e., if the object accelerates uniformly, then

a graph of velocity vs. clock time forms a straight line and, as a result
the average velocity of the object over a time interval is the average of its initial and final velocities for that time interval.

If velocity does not change at a constant rate this is usually not the case.

** For uniform acceleration, a(t) = a = constant, so

v(t) is an antiderivative v(t) = a t + c of the acceleration function
the arbitrary constant c is equal to v(0) so we can write v(t) = a t + v0

the position function is the antiderivative x(t) = .5 a t^2 + v0 t + c of the acceleration function
the artitrary constant c is equal to x(0) so we write x(t) = .5 a t^2 + v0 t + x0
the change in position `ds between clock times 0 and `dt is `ds = x(`dt) - x(0) = .5 a `dt^2 + v0 `dt + x0 - x0, so

`ds = v0 `dt + .5 a `dt^2.

If an object accelerates uniformly over a time interval then the average velocity is halfway between the initial and final velocities. It follows that the change from the initial to the average velocity is the same as the change from the average velocity to the final velocity. Initial, average and final velocities will be 'equally spaced'.

As a special case of the above, when a uniformly accelerating object is observed over a time interval in which it starts from rest, and only when it starts from rest, its average velocity will be half of its final velocity on that time interval; the final velocity will thus, under these special conditions, be double the average velocity.

Idea 5: Sequential Observations and Graphical Representation of Positions and Velocities

If the position of an object is observed a series of clock times, its average velocity `ds/ `dt over each time interval can be easily determined. From the resulting approximate average velocity vs. clock time information we can then estimate the approximate average rate `dv /` dt at which velocities change between midpoint times, giving us an approximate graph of acceleration vs. clock time.

If the velocity of an object is observed at a series of clock times, its approximate average velocity over each time interval is easily determined. The approximate distance the object moves over each time interval is then easily found. From the distances moved over the successive time intervals, the total change in the position of the object from the first clock time to any other clock time is then easily calculated.

Given a graph of velocity vs. clock time over a range of clock times, we can partition the graph by a series of time intervals.

For each time interval we can determine the approximate average velocity from the velocities at the beginning and end of the time interval. The approximate average velocity over an interval is the average 'altitude' of the corresponding trapezoid.
From the average velocities and the durations of the various time intervals, we can calculate the approximate displacement corresponding to each interval. Adding displacements to initial position we obtain position vs. clock time information and can use this information to construct an approximate position vs. time graph.
The displacement corresponding to each interval is represented by the area under the v vs. t graph which corresponds to that interval.
The displacement corresponding to an interval is approximated by the area of the corresponding trapezoid. The accumulated area through each trapezoid approximates the displacement from the initial clock time through the clock time at the end of that trapezoid.

Given a graph of position vs. clock time over a range of clock times, we can partition the graph by a series of time intervals.

The average slope between any two clock times represents the approximate velocity at the midpoint of those two clock times. If we make a table of average slope vs. midpoint clock time and construct the corresponding graph we obtain an approximate graph of velocity vs. clock time.

The velocity vs. time graph will represent the slopes of the resulting position vs. time graph; the position vs. time graph will represent the accumulated areas under the velocity vs. time graph. ** This observation is equivalent to the Fundamental Theorem of Calculus.

The position vs. clock time, velocity vs. clock time, and rate of velocity change vs.clock time (i.e., acceleration vs. clock time) chronicles can be represented by graphs, with the 'y' coordinate of each graph representing the rates of change, or slopes, of the preceding graph. (i.e., the velocity graph represents as 'y' coordinates the slopes of the position graph, the acceleration graph represents at 'y' coordinates the slopes of the velocity graph).

Idea 6: Newton's First Law and situations involving Uniform Acceleration

The rate at which the velocity of an object changes is called its acceleration. Acceleration is rate of change of velocity.

If an object is accelerating in the direction of its motion, it is speeding up. If it is accelerating in the direction opposite to that of its motion, it is slowing down.

For example, an automobile coasting freely on a south-facing slope accelerates to the south, whether it is traveling north or south. If it is traveling south, it travels with the slope and speeds up. If it is traveling north it travels against the slope and slows down. If it is standing still, it will in the next instant be traveling south and speeding up.

If an object accelerates perpendicular to its direction of motion, with zero acceleration in its direction of motion, then its direction of motion will change but its speed will not.

Newton's First Law observes that, in the absence of a net force, an object will not accelerate. An object which does not accelerate will change neither its speed nor its direction of motion.

In the vicinity of the surface of the Earth any freely falling object is observed to accelerate at very nearly the same constant rate, independent of where on Earth it is.

A falling object, or an object on a uniform incline, which accelerates freely without resistance accelerates uniformly. On a uniform incline whose slope as measured from the horizontal direction is small this acceleration is very nearly equal to the product of the acceleration of gravity and the slope.

An object accelerating freely, except for the influence of friction, on a uniform incline with small slope will have greater acceleration for greater slope. The change in the acceleration from one slope to another will be very nearly equal to the product of the acceleration of gravity and the difference in the ramp slopes. Thus the slope of a graph of acceleration vs. ramp slope will be very nearly equal to the acceleration of gravity.

When an object is in free fall near the surface of the Earth, with no external forces other than gravity acting on it, the net force on the object is vertical, with no horizontal component.

Its vertical motion is therefore characterized by a uniform acceleration equal to that of gravity, while its horizontal motion is characterized by zero acceleration, resulting in constant horizontal velocity.

Idea 7: Reasoning Out and Formulating Uniformly Accelerated Motion

We can organize our thinking about a problem by using simple 'flow diagrams' showing the 'flow' of our reasoning. These diagrams can be extremely useful in 'mapping out' our solution strategies on complex situations.

For uniformly accelerated motion in one direction, we can reason out the motion using the quantities `ds, `dt, v0, vf, a, `dv, and vAve, and in terms of the units of these quantities.

In order to think clearly about what is going on, we need to think in terms of all seven of these quantities.
We can, however, formulate uniformly accelerated motion in terms of any five of these variables, provided the five are independent.
The consensus formulation is in terms of `ds, `dt, v0, vf, and a.

To formulate uniformly accelerated motion in terms of `ds, `dt, v0, vf, and a, we begin by formulating the definitions of average acceleration and average velocity in terms of these five variables, obtaining the two equations `ds = (vf + v0) / 2 * `dt and vf = v0 + a * `dt.

If we eliminate vf from the two equations we obtain a third equation `ds = v0 `dt + .5 a `dt^2, and if we eliminate `dt from the two equations we obtain a fourth equation vf^2 = v0^2 + 2 a `ds.
Using these equations, given the values of three of the five variables we can solve for the remaining two without having to think much.
It is OK not to think too much about a situation provided we already understand situation thoroughly, as we do if we can reason out the basic situations without resorting to formulas, and if we can derive the formulas from the definitions.

Preliminary Assignment (This assignment should have been completed as part of the Initial Activities)

Introduction to Key Systems

· introductory_pendulum_experiment

· Using the Timer Program

· First 'Seed' Question

· Responses to Questions about Key Systems

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

1.1 – 1.7

initial timing experiment

View Physics material on GEN 1 CD and ponder questions posed in documentation.

text_00

ball on incline, rotating strap

B5. rates

query_00

Objectives:

Solve problems using the relationships among displacement, time interval and average velocity.
Accurately observe positions and clock times of events for a moving object.
Estimate and justify estimates of uncertainties related to preceding.
Use measurements and estimated uncertainties to estimate the uncertainty in the velocity of the object.
Use appropriate units throughout your calculations and perform the algebra of the units.

Technical statement of objectives:

0.01 Relate {`ds, `dt, vAve} `ds is the displacement of an object during an interval, `dt is the change in clock time during the interval and v_Ave is the average velocity during the interval. ('object' is no yet distinguished from 'particle') Introductory Problem Set 1 , hands-on activities

0.02 Measure the positions of objects and/or events, relative to a reference point and reference directions. Initial Timing Experiment

0.03 Estimate and justify estimates of uncertainties in measurement of positions and clock time. Initial Timing Experiment

0.04 Using measurements of position vs. clock time for two events determine the average velocity corresponding to the interval between the events, and based on uncertainty estimates determine the uncertainty in the result. Initial Timing Experiment

0.05 Include units at every step of every calculation for every quantity which has units, do the algebra of the units, compare units obtained with units expected at every step, where units errors exist identify and correct.

All objectives can be achieved using prerequisite knowledge of fractions and the algebra of fractional quantities. Universal common practice in all calculations in all queries, qa’s, class notes, text problems.

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

1.1

1.2

Formatting Guidelines and Conventions

Error Analysis I.

PHeT 1.26 Estimation

text_01

rubber bands

Memorize This Idea 1

B6. areas

PhET 1.16 Estimation

query_01

Objectives:

Format data reports as instructed. [formatting lab]
Calculate mean and standard deviation of a repeatedly measured quantity. [error analysis lab]
Know the definition of average rate. ['seed' question]
Apply the definition of average rate to the relationships among displacement, average velocity and time interval. ['seed' questions]
Know how observed value, uncertainty and percent uncertainty are related and be able to calculate any of these quantities from the values of the other two. [text]

1.01 Know class guidelines for formatting data using text format. Formatting Guidelines and Conventions

1.02 Compute and interpret within the context of an experiment the mean and standard deviation of a set of observed values. Error Analysis I

1.03 Know verbatim the definition of ‘average rate of change of A with respect to B’. ‘seed’ problem 1.1.

1.04 Apply the definition of average rate of change to identify the appropriate quantities and relate {`ds, vAve, `dt, average rate of change of position with respect to clock time}. ‘seed’ problem 1.1, Introductory Problem Set, qa_01

1.05 Relate {observed value of a quantity, uncertainty in observed value, percent uncertainty} Assumed general knowledge from elementary and secondary mathematics prerequisite to prerequisite courses; Text Chapter 1 also includes examples

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

qa_02

2.1 – 2.3

Fitting a Straight Line

Collaborative Labs Series 1

text_02

measure dominoes

#01

Rand Prob Asst 2 Prob 1

Memorize This Idea 2

B7. volumes

query_02

Objectives:

Reason out, based on the definitions of velocity and acceleration, the velocity and acceleration relationships for an interval.
Reason out uniformly accelerated motion on an interval based on the definitions of average velocity and average acceleration.
Represent the details of uniformly accelerated motion on an interval, using a trapezoidal v vs. t graph.
Using a hand sketch of data points make a reasonable estimate of the line of best fit and its equation.

Technical Statement of Objectives:

02.01 Relate {v0, vf, a_Ave, `dt, `dv, average rate of change of velocity with respect to clock time, average rate of change of position with respect to clock time, definition of average rate} based on the definition of average rate, where v0 and vf are the velocities of an object at the beginning and end of an interval, a_Ave is the average acceleration on the interval, `dt the change in clock time during the interval, `dv the change in velocity during the interval Class Notes, Introductory Problem Sets, ‘Seed’ problems ... related in context of ... fundamental systems etc ... and extend to arbitrary situations

02.02 Relate {v0, vf, a_Ave, `dt, `dv, `ds, v_Ave, average rate of change of velocity with respect to clock time, average rate of change of position with respect to clock time, definition of average rate} where `ds is the change in position, v_Ave is average velocity and other quantities are as previously defined Class Notes, Introductory Problem Sets, ‘Seed’ problems

02.03 Relate {v0, vf, a_Ave, `dt, `dv, `ds, v_Ave, definition of average rate, v vs. t ‘graph trapezoid’, ‘graph altitudes’, ‘trapezoid width’, slope, area} within the context of a graph in the v vs. t plane and the 'graph trapezoid' defined by v0, vf and `dt Class Notes, Introductory Problem Sets, ‘Seed’ problems

02.04 Hand-sketch a straight line to ‘best fit’ a set of y vs. t data, determine the slope of the sketched line, determine the equation of the sketched line. Fitting a Straight Line

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

qa_03

2.1

2.2

3.1

Measuring Distortion of Paper Rulers

121 alt: Brief Experiment: Paper Rulers)

text_03

velocity and `dy

#02-03

Rand Prob Asst 3

Rand Prob Week 2 Quiz 1

Memorize This Idea 3

B8. units of volume measure

query_03

Objectives:

… objectives are typically reinforced throughout the course, especially in subsequent assignments; listed at first encounter

Construct, properly label and interpret a trapezoidal approximation graph representing velocity vs. clock time information.
Use a trapezoidal approximation graph of velocity vs. clock time to infer approximate position vs. clock time.
Symbolically relate position, velocity, clock time and acceleration quantities for an interval of uniform acceleration, based on the properties of the 'graph trapezoid' representing v vs. t.
Relate position, velocity, acceleration and clock time quantities to the motion of a ball uniformly accelerating from rest down a ramp.
Know basic conversion factors for length and apply to convert lengths, areas and volumes.
Estimate and justify estimates of accuracy, precision and uncertainty for measurements made using rulers copied at various levels of reduction.
Symbolize the relationships among the quantities used to reason out and model uniform acceleration on an interval.

Technical Statements of Objectives:

03.01 Trapezoidal approximation graph: Relate {(t_i, y_i), segment [(t_i, 0) – (t_i, y_i) ) | 0 <= i <= n} U {slope_i, area_i, accumulatedArea_i, segment [(x_(i-1), y_(i-1) ) | 1 <= i <= n} U {rateOfSlopeChange_i, 1 <= i <= n-1 } U {graph labeling conventions}

03.02 Use a ‘trapezoidal approximation graph’ of v vs. t to infer a graph of position vs. t Class Notes, Introductory Problem Sets, ‘Seed’ problems

03.03 Relate and symbolize relationship: {v0, vf, a_Ave, `dt, `dv, `ds, v_Ave, definition of rate, v vs. t ‘graph trapezoid’, ‘graph altitudes’, ‘trapezoid width’, slope, area} Class Notes, Introductory Problem Sets, ‘Seed’ problems

03.04 Relate in context of a ball accelerating from rest down a ramp: {v0, vf, a_Ave, `dt, `dv, `ds, v_Ave, definition of rate, v vs. t ‘graph trapezoid’, ‘graph altitudes’, ‘trapezoid width’, slope, area} Class Notes, Introductory Problem Sets, ‘Seed’ problems

03.05 {SI units, metric prefixes, 1 inch = 2.54 cm, conversions of length, conversions of area, conversions of volume} General elementary and secondary knowledge, Text Ch 1, common practice in all materials, orientation exercises

Preliminary step {SI units, metric prefixes, 1 inch = 2.54 cm, conversions of length}

03.06 Accuracy and precision of measuring instruments Measuring Distortion of Paper Rulers, Text Ch 1

03.07 Relate and symbolize: {v0, vf, v_Ave, `dv, `dt, `ds, a, `ds, definitions of v_Ave and a_Ave}

Preliminary:
{`ds, `dt, v_Ave, v0, vf}
{`dv, `dt, a_Ave, v0, vf}

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

qa_04

4.1

Rotating Straw, Opposing Rubber Bands

PHeT 2.24 Moving Man

(text_04)

force on marble on incline

#04

Rand Prob Asst 4

query_04

Objectives:

Use position vs. clock time data to estimate behavior of velocity vs. clock time.
Analyze depth vs. clock time for water flowing from a uniform cylinder through a hole in the side of the cylinder.
Apply the relationships among angular displacement, time interval and average angular velocity.
Relate two sets of graphical length vs. length information representing three elastic objects.

04.01 Using measurements of position vs. clock time for a series of intervals, determine average velocity on each interval.

04.02 Flow experiment analysis: Relate {(t_i, y_i), segment [(t_i, 0) – (t_i, y_i) ) | 0 <= i <= n} U {slope_i, area_i, accumulatedArea_i, segment [(x_(i-1), y_(i-1) ) | 1 <= i <= n} U {rateOfSlopeChange_i, 1 <= i <= n-1 } U {graph, graph labeling conventions} -> {(t_mid_i, v_Ave_i) | 1 <= i <= n} -> {s(t) = A t^2 + B t + C} where t_i is the ith clock time and y_i the ith vertical position observed in the flow experiment, t_mid_i the clock time at the midpoint of the ith interval class notes

04.03 Relate {`dtheta, `dt, omega_Ave} rotating straw, opposing rubber bands

04.04 Relate {(L_a_i, L_b_i) | 1 <= i <= n} U {(L_a_i, L_c_i) | 1 <= i <= n} U {rate_ba, rate_ca, rate_cb} where L_a_i, L_b_i, L_c_i are lengths of opposing elastic objects, rate_xy is the rate at which elastic object x changes length with respect to change in length of elastic object y } rotating straw, opposing rubber bands

Preliminary:

Graph L_b vs. L_a

Graph L_c vs. L_a

Find slopes of graphs.

Intepret slopes as average rates.

Infer graph of L_c vs. L_b.

Asst

seed

Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

qa_05

5.1

2.4 – 2.6

The Pearl Pendulum

Timing a Ball down a Ramp

text_05

buoyant balance

#05-06

Memorize This Idea 4

synopsis of principles and common errors in analysis of motion

synopsis: Understand ideas 1-3

Rand Prob week 3 quiz 1

query_05

Objectives:

Apply 'flow diagrams' to relate the quantities used to analyze uniformly accelerated motion.
Calculate and relate accelerations of an object coasting down ramps with various small slopes to infer acceleration vs. ramp slope.
Estimate uncertainties in the preceding.
Observe and quantify details of the periodic motion of a pendulum.
Estimate the minimal difference in slope detectable by timing a ball down a ramp using the TIMER program.
Reason out the first two equations of uniformly accelerated motion using the definitions of average velocity and average acceleration.

Technical Statements:

05.01 Use ‘flow diagrams’ to relate {`ds, `dt, v_Ave, `dv, v0, vf, a} Class Notes

05.02 Relate {(`dt_i, h_i, L_i,) | 1 <= i <= n} U { a_i, rampSlope_i | 1 <= i <= n} U {graphSlope}, where `dt_i is the time required to accelerate down a ramp of length L_i one of whose ends is elevated a vertical distance h_i above the other, a_i is the acceleration on that ramp, rampSlope_i is the slope of the ramp relative to horizontal, and graphSlope is the slope of the graph of the a_i vs. rampSlope_i. Class Notes, Timing a Ball Down a Ramp

Preliminary:

{`dt_i, L_i, a_i}

{L_i, h_i, rampSlope_i}

05.03 Estimate the uncertainties in graphSlope in terms of the uncertainties in observations of {(`dt_i, h_i, L_i,) | 1 <= i <= n}. Timing a Ball Down a Ramp

05.04 Observe and quantify the periodic behavior of a pendulum by measuring, relative to time of release, clock times at which it returns to its equilibrium point and associated time intervals. The Pearl Pendulum

05.05 Time a ball down a ramp at various slopes. Determine which slope increments result in observable differences in the results, speculate on the minimal slope increment for which timing with the TIMER program could distinguish the change. Timing a Ball Down a Ramp, Class Notes

05.06 Reason out the first two equations of uniformly accelerated motion by drawing all possible conclusions from given v0, vf, `dt, and depict the reasoning using flow diagrams. Class Notes, qa_05

Asst

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Introductory Problem Sets

lab/activity

text

streamline

Class Notes

other

query

qa_06

6.1

Rubber Band Calibrations

text_06

Units, Meaurements, Uncertainties

(University Physics also includes Vectors)

colliding marbles

#07

Memorize This Ideas 5 and 6

Rand Prob week 2 quiz 2

synopsis: Understand ideas 4, 5, 6, 7

query_06

Objectives:

Correctly identify interval(s) useful in analyzing a situation, and the quantities used to reason out and analyze uniformly accelerated motion on each interval.
Obtain and apply force vs. length information for elastic objects.
Derive the four equations of uniformly accelerated motion.
Identify those uniform-acceleration situations which can be reasoned out using only definitions, without the use of the equations of motion, and which cannot.
Apply the equations of uniformly accelerated motion to analyze motion for intervals of uniformly accelerated motion.

Technical Statements:

06.01 Identify appropriate interval(s) and the quantities v0, vf, `dt, a, `ds, aAve, `dv, v_Ave for each interval, when they appear in problems or in hands-on situations. ‘seed’ problem

06.02 Observe and record force vs. length characteristics for rubber bands, use results to predict length for given force and force for given length rubber band calibrations

06.03 Derive the first two equations of motion by analyzing the implications of v0, vf and `dt; derivate the third and fourth equations by eliminating v_f and `dt, respectively, from the first two class notes

06.04 List the combinations of three of the five quantities v0, vf, `dt, `ds and a, and identify which of the 10 possible combinations can be reasoned out using only the definitions of average velocity and average acceleration.

06.05 For any uniform-acceleration situation in which three of the five quantities v0, vf, `dt, `ds and a are known, be able to use the equations to find the other two. Qa_06

Asst

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Introductory Problem Sets

lab/activity

text

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Class Notes

other

query

qa_07

7.1

7.2

Error Analysis Part II, Data Program

PHeT 3.51 Motion in 2D

text_07

Uniformly Accelerated Motion in 1 dimension

measuring g

Memorize This Idea 7

Rand Prob week 3 quiz

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Objectives:

Measure the acceleration of gravity.
Use technology to calculate difference quotients between multiple ordered data points.

Technical Statements:

07.01 Measure the acceleration of gravity, establish that this quantity is independent of mass qa_07

07.02 Using technology calculate {( (y_i – y_(i-1)) / (t_i – t(i-1) ), t_mid_i = (t_(i-1) + t_i) / 2 | 1 <= i <= n} when given {(t_i, y_i) | 0 <= i <= n}. Error Analysis Part II, Data Program

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8.1

8.2

Hypothesis Testing with Time Intervals

PHeT 3.43 Projectile Motion

(text_08)

torque and rotation

Rand Prob week 4 quiz 1

Practice Major Quiz (qa)

query_08

Objectives:

Analyze the motion of freely falling objects.

Technical Statement:

08.01 Know and apply the acceleration of gravity to the uniformly accelerated motion of objects in vertical free fall. qa_08

Module 2: Assignments 9 - 20

Test #1 (cumulative through Module 2) assigned as part of Assignment 21

Summary of Main Ideas in Module 2

Having completed and reviewed your work in the assignments comprising this Module you should understand these ideas and be able to apply them to solving problems and analyzing real-world situations.

Idea 8: Newton's Laws of Motion

Newton's First Law tells us that whenever the net force on an object is 0, the magnitude and direction of the velocity of the object will not change.

We therefore say that any object either at rest or in motion at constant velocity with respect to some fixed direction is in equilibrium with respect to that direction.

From Newton's First Law we conclude some useful things:

When an object is either at rest or moving along a constant incline, then since it is not moving in the direction perpendicular to the incline it must be in equilibrium with respect to that perpendicular direction.
Hence the sum of all forces perpendicular to the incline must be 0.

When an object is suspended at rest or suspended and moving at constant velocity by a number of ropes, chains, rubber bands, cables, or similar devices, the sum of all the forces exerted on the object in any direction must be zero.

If all the suspending devices lie in a plane, it is possible to express all the forces on the object as the sum of their components in the vertical and horizontal (y and x) directions.

Newton's Third Law tells us that for every force exerted on an object by some source the object exerts an equal and opposite force on the source.

From Newton's Third Law we conclude some useful things:

If we push on an object while standing on a frictionless skateboard, the object will exert an equal and opposite force on us, and our mass plus that of the skateboard will have the acceleration determined by the magnitude of this force and our mass, and the direction of this force.

If a string exerts a tension force on a suspended mass, then the mass exerts an equal and opposite force on the string.

If a frictional force resists the motion of an object as it slides across some surface, then the object exerts a force equal and opposite to the frictional force, and therefore in the direction of the object's motion.

As the tires on your car attempt to spin along the road surface in some direction, they exert a frictional force on the road and the road exerts an equal and opposite force on the tires. It is this frictional force exerted on the tires that accelerates the automobile.

Using Newton's Third Law and similar principles we can sketch and analyze the forces acting on any system we care to define.

Newton's Second Law tells us that the net force Fnet required to accelerate a mass m at rate a is Fnet = m * a.

Standard situations include:

Atwood machine, where Fnet = m1 g - m2 g and mass = m1 + m2. String tension is a force internal to this system so does not contribute to the net force.
Person standing on a scale in an accelerating elevator, where Fnet = Fscale - m g and Fnet = m a so Fscale - m g = m a.
Mass m1 on level frictionless table attached by cord to mass m2 hanging over light frictionless pulley, where Fnet = m2 g acts on system with mass is m = m1 + m2. String tension is a force internal to this system so does not contribute to the net force.
Pendulum with mass m, length L pulled back distance x from equilibrium and released, where Fnet = m g / L * x directed toward equilibrium position.
Object of mass m moving freely on frictionless incline at angle theta, where Fnet = m g * cos(270 deg + theta) = m g * sin(theta) is the component of the weight parallel to the incline, directed down the incline. The normal force is exerted perpendicular to the incline and balances the perpendicular component of the weight.

Idea 9: Work (net force applied through displacement) and the Work-Energy Theorem

We compare the effect of applying a net force F to a mass m

while the mass moves through a known displacement `ds with the effect of a
pplying the same net force to the same mass for a time interval `dt.

If we apply a net force Fnet to a mass m while that mass moves through the displacement `ds in the direction of the force, the object's velocity in that direction changes at rate a = F / m.

The effect of this force on the velocity of the object is found by substituting a = F / m into the equation vf^2 = v0^2 + 2 a `ds.

We obtain the equation

Fnet * `ds = .5 m vf^2 - .5 m v0^2.

This tells us that the product Fnet * `ds gives us the change in the quantity .5 m v^2.

We define kinetic energy, or energy of motion, as the quantity

KE = .5 m v^2.

With this definition the difference .5 m vf^2 - .5 m v0^2 is expressed as KEf - KE0 = `dKE, and we have

Fnet * `ds = `dKE.

The quantity F * `ds, where `ds is displacement in the direction of the force F, is called the work done by the force F.

So our statement that Fnet * `ds = `dKE can be restated as

work by net force on system = change in kinetic energy.

The above statement is the most rudimentary form of the Work-Energy Theorem:

Work-Energy Theorem: work by net force on system = change in kinetic energy.

We can observe that by Newton's Third Law the work done by the net force on a system must be equal and opposite to the work done by the system against that net force, so that

work done by system = -(work done on system) = -`dKE,

and that therefore

work done by system against net force on system + `dKE = 0, or
`dWby + `dKE = 0.

If it is understood that `dW stands for the work done BY the system and `dKE for the KE change of the system we have

Work-Energy Theorem (first form): `dW + `dKE = 0 (note that `dW is work done BY the system)

** Using calculus we can generalize the Work-Energy Theorem to say that the integral of (net force on syste) * (displacement in the direction of that force) is equal to the change in the kinetic energy of the object which the force is applied:

** int( F ds, s, a, b) = `dKE,

** where the force is applied from displacement s = a to s = b and `dKE is the change in the kinetic energy of the object between these two displacements.

** This integral corresponds to the area under the force vs. displacement curve.

In all cases it is understood that the displacement is the displacement in the direction of the force.

Idea 10: Impulse (force applied over time interval) and Momentum; Conservation of Momentum

In contrast to the situation where we apply a net force through a give displcacement, here we apply the net force over a given time interval.

If we apply a net force Fnet to a mass m for a time interval `dt, the effect of the force is found by substituting a = F / m into the equation vf = v0 + a * `dt.

We obtain the equation

Fnet * `dt = m vf - m v0.

We call the quantity mv the momentum of the object and the quantity Fnet * `dt the impulse of the force.

The statement that Fnet * `dt = m vf - m v0 can therefore be restated as

impulse = change in momentum.

This is called the Impulse-Momentum Theorem.

** The Impulse-Momentum Theorem is easily generalized using calculus to include situations in which either or both the mass and velocity of the object changes while the force is applied.
** In this form the Impulse-Momentum Theorem reads

** F dt = dp,

** where p = mv stands for momentum.
** A simple rearrangement gives us

** F = dp / dt: force is the instantaneous rate of change of momentum.

By Newton's Third Law the forces exerted by two interacting objects one one another are equal and opposite. It follows that the impulses applied by the object to one another are equal and opposite. Hence the net change in momentum is equal and opposite. This idea is easily extended to any closed system of objects--i.e., any system in which the only forces exerted on the objects in the system are exerted by other objects (and in fact to any system where the vector sum of all the forces exerted on the system is zero). Impulses in such a system are applied in the form of equal and opposite forces, resulting in equal and opposite momentum changes. The result is that the net momentum change of the system must always be zero, so that the total momentum of the system remains constant.

Conservation of momentum: The total momentum of any closed system is unchanging.

Idea 11: The Work-Energy Theorem can be rephrased in terms of conservative and nonconservative forces

The Work-Energy Theorem can be restated in terms of conservative and nonconservative forces, introducing the idea of potential energy.

A conservative force is one that stores energy in such a way that we can recover the work done against the force.

For example a gravitational force is conservative since the kinetic energy we obtain from dropping an object from a certain height, in the ideal situation in which air resistance and other nongravitational forces are eliminated, is equal to the work that must be done against gravity to raise the object to that height.
This occurs because the gravitational force acts as the object is raised, requiring an equal and opposite force to raise the object at constant velocity, and then acts as the object falls through the same distances.

The elastic forces exerted by a spring or a rubber band as it is stretched are in the ideal case conservative, since they are experienced again in the opposite direction upon release (this is not quite the case for real rubber band and springs, which have some internal friction both when being stretched and released so that in fact more work must be done in stretching than is done in the 'snap back'; however we take it to be so for 'ideal' springs and rubber bands, in which the frictional forces are considered to be negligible) so that the work done against the elastic force can be recovered.

The work done against a conservative force is therefore considered to be 'stored' in the system until it is released; upon release it can perform work on another system and/or increase the kinetic energy of the system.

We therefore say that the work done by a system against a conservative force increases the 'potential energy' of the system, and that upon release potential energy is converted into some other form of work and/or energy.

A nonconservative force is one from which we can get back none of the work done against this force.

For example work done against friction typically increases the random kinetic energy (thermal energy) of objects which are being moved across one another.
This energy is random and diffuses away from its points of origin.
While theoretically possible it is statistically impossible for such energy to reorganize itself and do useful work.

Recall the Work-Energy Theorem:

Work-Energy Theorem (first form): `dW + `dKE = 0 (note that `dW is work done BY the system AGAINST the net force acting on the system)

If we denote by `dWnoncons only the work done against nonconservative forces in a system and by `dPE the work done by the system against conservative forces, then `dW = `dWnoncons + `dWcons and the Work-Energy Theorem can be restated as follows:

Work-Energy Theorem (second form): `dWnoncons + `dPE + `dKE = 0

The system can do positive work `dWnoncons against nonconservative forces only by decreasing the sum of its potential and kinetic energy.
If no external forces act on it, then `dW = 0 so that the system can increase its potential energy only at the expense of an equal amount of its kinetic energy (remember that `dW cannot be negative), and can increase its kinetic energy only at the expense of an equal amount of its potential energy.

Idea 12: Vector Quantites

A vector quantity is one that has magnitude and direction. Examples include velocity, acceleration, force and momentum.

A vector in two dimensions can be represented on a two-dimensional coordinate system with an x and a y axis.
If we measure the angle theta of the vector in the counterclockwise direction from the positive x axis then the action of the vector is completely equivalent to the combined action of its x and y components V cos(theta) and V sin(theta), respectively, where V is the magnitude of the vector.
To add two or more vectors we can proceed as follows:

Add the x components of all the vectors to get the x component Rx of the resultant vector.

Add the y components of all the vectors to get the y component Ry of the resultant vector.

The magnitude of the resultant vector is R = sqrt(Rx^2 + Ry^2) and the angle is theta = arctan(Ry / Rx), provided Rx is positive, or arctan(Ry / Rx) + 180 degrees if Rx is negative.

Vector quantities have magnitude and direction and include, among many others, displacement, velocity, acceleration, force, momentum and gravitational field.

The following, among many others, have no direction and therefore not vector quantities: distance, time, mass, volume, density, energy.

Important applications of vectors to forces include the following:

The weight w of an object on a incline which makes angle theta with horizontal acts in the vertical downward direction; if the xy coordinate system is oriented so the x axis is directed parallel to the incline with the y axis perpendicular, then the weight is at angle 270 deg - theta as measured counterclockwise from the positive x axis, so it has x and y components wParallel = w cos(270 deg - theta) parallel to the incline (i.e., in the x direction), and wPerpendicular = w sin(270deg - theta) perpendicular to the incline (i.e., in the y direction). [Since | cos(270 deg - theta) | = | sin(theta) | and | sin(270 deg - theta) | = | cos(theta) |, these formulas are consistent with those obtained using right-triangle trigonometry. Your text likely uses right-triangle trigonometry rather than the circular-model definitions used in most of this course. The two approaches give the same results, but the pictures are slightly different.]
The tension T in a pendulum is, for displacements from equilibrium which are much less than the length of the pendulum, very nearly equal to the weight of the pendulum. If the angle of the pendulum from vertical is theta, then tension vector makes angle 90 deg + theta as measured counterclockwise from the positive x axis. This vector therefore has component w cos(90 deg + theta) in the x direction, pointing back toward the equilibrium position. Since the displacement from equilibrium is x = L cos(90 deg + theta), it follows that cos(90 deg + theta) = x / L so that the x component of the tension is Tx = x * w/ L. That is, the tension component tending to restore the pendulum to equilibrium is proportional to the displacement from equilibrium. Note also that the horizontal tension is in the same proportion to pendulum weight as displacement x is to pendulum length. [Note that right-angle trigonometry can also be used to analyze these forces, with the result that the horizontal component of tension has magnitude w sin( | theta | ). Since | cos(90 deg + theta) | = | sin(theta) | this is equivalent to the result obtained using the circular model. ]

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9.1

3.1-3.6

Analysis of Initial Ball Down Ramp Data

Deterioration of Difference Quotients

(text_09)

#08-09

query_09

Objectives:

Be able to state and apply the definition of weight. [ qa ]
Apply the concept of propagation of errors. [seed]
Calculate and/or explain the propagation of errors when calculating successive difference quotients. [lab]
Apply the definition of net force and its relationship with mass and acceleration. [introductory problem set]
Apply the definition of work. [introductory problem set]
Describe an experiment to determine whether acceleration is constant on a constant incline. [class notes]
Describe an experiment to test whether the acceleration of a given mass is proportional to the net force on that mass. [class notes]
Describe an experiment to test whether the change in v^2 is proportional to the change in net force * displacement. [class notes]

09.01 Relate {F_grav, weight, F_net, m, a, g} where F_grav is the force exerted by gravity on mass m, a is the acceleration of the mass, g the acceleration of gravity, weight is the weight of the mass, F_net the net force acting on the mass qa_09

Preliminary {F_grav, m, g}

09.02 Explain how uncertainties in measurement of position and timing tend to be multiplied in calculations of velocity, which errors are then multiplied once more in calculations of acceleration. Deterioration of Difference Quotients

09.03 Demonstrate how curve fitting can mitigate the multiplication of errors inherent in calculations by successive difference quotients. Deterioration of Difference Quotients

09.04 Relate {F_net, a, m, `dW, `ds} where `dW is the work done on the mass by the net force, `ds the displacement of the mass, and other quantities are as defined previously Class Notes

Preliminary {F_net, `dW, `ds}

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10.1

3.7 – 3.12

Hypothesis Testing with Data from Pearl Pendulum and Ball Down Ramp

text_10

Rand Prob week 5 quiz 1

Rand Prob week 4 quiz 2

Physics I Basic Quantities and Relationships lines 10, 11, 12

synopsis: Understand idea 8

query_10

Objectives:

Calculate and apply means and standard deviations. [lab]
Be able to state and apply Newton's Second Law. [ introductory problem sets, qa ]
Sum individual forces to get the net force on a mass and apply to the analysis of its motion. [qa]
Relate the product a `ds to change in v^2, and F_net `ds to change in m v^2. [introductory problem sets]
Explain how to experimentally test the proportionalities implicit in the preceding. [previous class notes]

10.01 Test the hypothesis that two sets of observations of a similar quantity under different conditions indicate a difference due to the change in conditions, using the criterion that the corresponding intervals (mean - 1 standard deviation, mean +_ 1 standard deviation) do not overlap. Hypothesis testing with data from Pearl Pendulum and Ball Down Ramp

10.02 Relate {F_net, m, a, Newton’s Second Law} qa_10, Introductory Problem Sets

10.03 Relate {F_i, 1 <= i <= n} U {F_net, m, a} U {v0, vf, `dv, vAve, `ds, `dt} where F_i is the ith force acting on the point mass m qa_10, Introductory Problem Set

10.04 Explain why a * `ds should be proportional to the change in v^2, according to the fourth equation of uniformly accelerated motion; and why it follows that F_net * `ds should be proportional to the change in m v^2. Class Notes

10.05 Explain how to experimentally demonstrate that a `ds is proportional to change in v^2, and that F_net * `ds is proportional to the change in m v^2. Class Notes

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11.1

3.15 – 3.19

Angular Velocity of a Strap

text_11

#10-11

Rand Prob week 5 quiz 2

query_11

Objectives:

Determine whether a given force acts on an object or is exerted by the object. [introductory problem set, class notes]
Determine whether the work done by the force in the preceding does positive or negative work. [qa, introductory problem set, class notes]
Derive the work-kinetic energy theorem for constant force exerted on constant mass. [introductory problem set]
Apply the work-kinetic energy theorem. [introductory problem set, class notes]
Observe position vs. clock time for a rotating object [lab ]
Obtain or explain how to obtain experimental evidence that the net force on an object of constant mass is proportional to its acceleration.

11.01 Distinguish forces acting on a system or object from forces exerted by the system or object.

11.02 Distinguish work done on a system or object from work done by the system or object.

11.03 Show how the work-kinetic energy theorem follows from the fourth equation of uniformly accelerated motion and Newton’s Second Law

11.04 Relate {`dW_net, F_net, `ds, `dKE} for an object on a given interval, where `dKE is the change in the KE of the object.

Preliminary {`dW_net, `dKE}

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12.1

3.13-3.14

Ball and Ramp Projectile Behavior

text_12

#12

synopsis: Understand idea 9

query_12

Objectives:

Analyze the motion and energy changes of an Atwood machine for a given interval. [introductory problem sets, qa, seed]
Determine the parallel component of a given mass on a given small slope in the absence of friction. [qa]
Determine the parallel component of a given mass on a given small slope in the presence of friction. [qa]
Analyze the motion of a projectile. [lab]
Analyze forces exerted by and energy changes of an elastic object. [class notes]

12.01 Explain how a cart with a number of small weights, a pulley and a hanger can be used to validate that F_net is proportional to acceleration, for a system of constant mass. qa_12, Introset

12.02 Relate {m_1, m_2, fFrict, F_net, F_grav, a} U {`ds, `dt, v0, vf, v_Ave, `dv, `dW_frict, `dW_net, `dKE, `dPE} for two masses suspended by a light string over a light pulley. qa_12, Introset

Preliminary

{m_1, m_2, F_net} for f_frict = 0

{m_1, m_2, F_net, a} for f_frict = 0

{m_1, m_2, f_Frict, F_net}

{m_1, m_2, f_Frict, F_net, a}

{m_1, m_2, `ds, `dW_grav, `dPE}

{m_1, m_2, F_net, `dKE}

12.03 Relate {weight_parallel, slope, m, g, weight} for a mass m on a small slope, with weight_parallel being the component of the weight of the mass parallel to the incline qa_12

Preliminary: {weight, slope, weight_parallel}

12.04 Relate {weight_parallel, slope, m, g, mu, F_net, F_frict, a_up, a_down} for a mass on a small slope, where mu is the applicable coefficient of friction, F_frict the frictional force, a_up and a_down the acceleration when the mass is moving up or down the slope, mu the applicable coefficient of friction qa_12

Preliminary:

{m, g, mu, F_frict}

{F_frict, weight_parallel, F_net, a_up, a_down, m}

12.05 Relate {`ds_y, v0_y, vf_y, `dt} U {v_x, `ds_x, `dt} for a projectile, especially in the case where v0_y is negligible and `ds_y, `ds_x are known Ball and Ramp projectile behavior

Preliminary:

{`ds_y, `dt, vf_y} when v0_y = 0

{`ds_x, `dt, v_x}

12.06 Using a graph of tension vs. length for an elastic object: {F_tension(L), L, (a, b), F_tension(a), F_tension(b), F_ave_ab, `dW_ab} where F_tension(x) is the tension in a light elastic object at length L, L its length, F_ave_ab is the average tension between lengths a and b, `dW_ab the magnitude of the work done by the tension force as the object is stretched from length a to length b

Preliminary:

{a, b, F_tension(a), F_tension(b), F_ave_ab} using a graph of F_tension(x) vs. x

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13.1

3.22 – 3.28

Uniformity of Acceleration for a Ball on a Ramp

Analysis of Data from Angular Velocity of a Strap

text_13

Force and acceleration, forces in 1 dimension

#13

Memorize This: Idea 8

query_13

Objectives:

Apply the work-kinetic energy theorem. [qa, introductory problem sets]
Apply the work-energy theorem. [qa, introductory problem sets]
Experimentally test uniformity of acceleration on an incline. [lab]
Experimentally test uniformity of angular acceleration of a rotating strap. [lab]
Apply Newton's Second Law to the motion of a particle subject to various forces. [ ]

13.01 Relate {`dW_net_on, F_ave_net, `ds, `dKE} where `ds is the displacement of an object during an interval, `dW_net_on the work done on the object by the net force, F_ave_net the average net force exerted on the object and `dKE the change in its kinetic energy qa_13

13.02 Relate {`dW_on, `dW_by, `dKE, `dPE, `dW_net_on, `dW_net_by, `dW_cons_on, `dW_cons_by} where F_ave is the average value of a force exerted on the object, `dW_on the work done by the force F_ave acting on the object, `dW_by the work done by the object against the net force by the object, `dPE the change in the potential energy of the object, `dW_cons_on the work done by conservative forces acting on the object, `dW_cons_by the work done by the object against conservative forces, and other quantities are as previously defined introset

Preliminary:

{`dW_on, `dW_by}
{`dW_on, `dW_cons_on, `dKE, `dPE}

13.03 Test uniformity of acceleration for a ball coasting from rest on a constant incline. Uniformity of Acceleration for a Ball on a Ramp

13.04 Test uniformity of angular acceleration for a rotating strap coasting to rest; test hypothesis that frictional torque is independent of angular velocity. Analysis of Data from Angular Velocity of a Strap

13.05 Apply Newton's Second Law to the motion of a particle subject to various forces. eg fish line, bicycle and rider

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14.1

5.1 – 5.5

text_14

Force and acceleration, forces in 1 dimension

#14 - 15

query_14

Objectives:

Analyze forces, work, kinetic energy and potential energy for an object on an incline. [qa, class notes]
State how pendulum mass, length, position with respect to equilibrium and net force are related. [class notes]
Demonstrate how the components of a vector are related to its magnitude and direction. [introductory problem sets]
Demonstrate how the components, magnitude and direction of the sum of two vectors is related to the components, magnitudes and directions of those vectors. [introductory problem sets]

14.01 Relate {`dW_on, `dW_by, `dKE, `dPE, F_grav, `ds, KE_0, KE_f, F_frict, `dy, m} for an automobile on an incline during an interval on which its displacement on the incline is `ds, where KE_0 and KE_f are the automobile's kinetic energy at the beginning and end of the interval, m its mass, F_frict the frictional force acting on it, `dy the change in its vertical position qa_14

14.02 Relate {m, L, F_net, x} for a simple pendulum of mass m, where F_net is the net force and x its position with respect to equilibrium class notes

14.03 Relate {R, theta, R_x, R_y} where R and theta are the magnitude and angle (as measured counterclockwise from the positive x axis), R_x the x component and R_y the y component of a vector

14.04 Relate {R_1, theta_1, R_1_x, R_1_y, R_2, theta_2, R_2_x, R_2_y, R, theta, R_x, R_y}, where R_1 and R_2 are vectors and R the vector sum of R_1 and R_2.

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15.1

4.1 – 4.7

Experiment 7, Measuring masses: view only (on DVD)

text_15

Work, kinetic energy, potential energy

Physics I Basic Quantities and Relationships lines 25-27, 35-51

synopsis: Understand idea 10

query_15

Objectives:

State the impulse-momentum theorem. [qa, introductory problem set]
Apply the impulse-momentum theorem to solve problems. [qa, introductory problem set]
Derive the impulse-momentum theorem. [qa, introductory problem set]

15.01 Relate {F_net, `dt, `d(m v), m, `dv, `dp} U {a_Ave, F_ave, v0, vf, `dv} qa_15

15.02 Explain how Newton’s Second Law and the second equation of motion yield the impulse-momentum theorem. qa_15

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16.1

Force Vectors

text_16

Conservation of Energy

synopsis: Understand idea 11

query_16

Objectives:

Analyze the motion a projectile whose initial velocity is horizontal. [qa]
Analyze the resultant of the tension forces exerted by three or more rubber bands on a common point. [seed]
Find the resultant of three forces not all directed along the same line. [lab]
Demonstrate that the resultant force on an object in equilibrium is zero. [lab]

16.01 Relate {`ds_y, `dt} U {v_x, `ds_x, `dt} for a projectile whose initial velocity in the vertical direction is zero, where _x and _y denote x and y components of vector quantities qa_16 needed for asst 12 on projectile behavior of ball

16.02 Relate {(x_init_i, y_init_i), (x_term_i, y_term_i), L_i, F_i, theta_i, F_i_x, F_i_y, graph of F_i vs. L_i | 1 <= i <= n} U {F_net_x, F_net_y, theta, F_net} for a combination of linear elastic objects exerting forces on a common point, where L_i and F_i are the length and tension of the ith object, F_net_x and F_net_x the x and y components of the net force exerted on the common point by all the objects, and x_init and x_term are x coordinates of the two ends of an object and y_init and y_term the corresponding y components of the ends. Forces lab

Preliminary:

{(x_init_i, y_init_i), (x_term_i, y_term_i), L_i, F_i}
{(x_init_i, y_init_i), (x_term_i, y_term_i), L_i, theta_i}

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17.1

17.2

4.8 – 4.12

CD labeled Ph1 Sets 4, 5, 6

Force vs displacement 1

text_17

Conservation of Energy

#16 - 18

synopsis: Understand idea 12

Memorize This: Idea 11

query_17

Objectives:

Analyze elastic and/or inelastic collisions of two masses moving along a straight line. [qa, introductory problem sets]
Analyze the conact forces during collision in an elastic and/or inelastic collisions of two masses moving along a straight line. [qa, introductory problem sets]
Apply momentum conservation to a collision of two objects moving along a straight line. [introductory problem sets]
Analyze tension vs. length data and associated energy changes for an elastic object. [lab]

17.01 Relate {m_1, m_2, v_1, v_2, v_1’, v_2’} U {elastic, inelastic}, where m_1 and m_2 are the masses of two objects with respective velocities v_1 and v_2 comprising an isolated system, which collide either elastically or inelastically (as specified) and move apart with respective velocities v_1 ' and v_2 ' qa_17, introset

17.02 Relate {m_1, m_2, v_1, v_2, v_1’, v_2’, F_ave, `dt, `dv_1, `dv_2, a_Ave_1, a_Ave_2, `dp_1, `dp_2}, where F_ave is the magnitude of the average force (more specifically time-averaged force) between the objects during the time interval `dt during which they interact, a_Ave_i is the average acceleration of the ith object, `dp is change in momentum, and other quantities are as previously defined.

Preliminary:

{F_ave, |`dp_1|, |`dp_2|, `dt}
{m_1, m_2, a_Ave_1, a_Ave_2}

17.03 Relate {tension_i, length_i | 1 <= i <= n} U {(a, b), `dW_on_ab, `dPE_ab} assuming tension forces to be conservative, where tension_i is the tension exerted by an elastic object when its length is length_i, (a, b) is an interval of length, `dW_on_ab the work done on the elastic object as its length changes from a to b, `dPE_ab is the corresponding change in the elastic potential energy of the object, and all quantities are reasonably approximated force vs. displacement 1

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18.1

18.2

4.13 – 4.20

Energy Conversion 1

text_18

query_18

Objectives:

Analyze the 'rubber band and rail' experiment. [lab]
Analyze collisions of two objects in two dimensions. [class notes, qa]

Objectives:

18.01 Relate {F_tension_ave, `dL, `dW_rb_on, `dW_frict, `dPE, `ds, F_frict_by, `dKE} for ‘rail’ experiment, where `dW_rb is the work done by the rubber band on the 'rail', `dPE the potential energy change of the rubber band, F_tension_ave the average tension of the rubber band during the interval of contact with the moving 'rail', `dL the change in the length during the interval, `dW_rb the work done on the 'rail' by the rubber band, `ds the displacement of the 'rail' along the surface on which it slides, F_frict the frictional force between the 'rail' and the surface, `dW_frict_by the work done by the rail against friction, `dKE the change in kinetic energy during the interval of contact with the rubber band energy conversion 1

Preliminary:

{`ds, F_frict, `dW_frict}

{F_tension_ave, `dL, `dW_rb_on, `dPE}

18.02 Relate {m_1, m_2, v_1_x, v_2_x, v_1_y, v_2_y, v_1_x ‘, v_2_x ‘, v_1_y ‘, v_2_y ‘} U {elastic, inelastic}, where _x and _y denote the x and y components of the vector quantities, for an isolated system of two masses

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19.1

19.2

19.3

text_19

Conservation of Momentum, Impulse-momentum

#19 - 21

Physics I Basic Quantities and Relationships lines 28-34

Memorize This: Idea 10

query_19

Objectives:

Analyze the forces on a mass resting on an incline of arbitrary angle of elevation. [qa, seed, class notes]
Explain how a force is equivalent to its components. [qa, seed, class notes]
Analyze the motion of a projectile whose initial velocity is not known to be horizontal. [qa, seed, class notes]

19.01 Relate {m, alpha, theta, g, F_grav_parallel, F_grav_perpendicular, coordinate system}, where alpha is the angle of elevation of the incline, the y axis of the coordinate system is perpendicular to the incline and is directed to that a vector in the upward vertical direction has a positive y component, and theta the angle of the weight of the mass m with respect to the positive x axis of the coordinate system qa_19

19.02 Explain how a force is equivalent to its components. Class notes?

19.03 Relate {v0, theta_0, y_0, `ds_y, `ds_x, `dt, v0_x, v0_y} for a projectile, where theta_0 is the initial angle of elevation, y_0 the initial vertical position, and subscripts _x and _y indicate vector components parallel to the coordinate axes

Preliminary:

{`ds_y, v0, theta_0, `dt}

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20.1

5.6 – 5.11

PHeT 9.22 Balancing Act

Torque Experiment

text_20

Conservation of Momentum, Impulse-momentum

Physics I Practice Test 1

query_20

Go to Tests and investigate Test #1. Note that the Physics 121 test covers Introductory Problem Sets 3 and 4 and the experiments done between Assignments 10 and 19, inclusive. Physics 201, 231 and 241 tests cover all material from the first assignment through Assignment 19, but with concentration on Assignments 10 - 19.

Objectives:

Analyze the motion of a mass on an incline attached to a weight suspended over a pulley. [qa]
Find the torque of a set of forces relative to a given axis of rotation. [lab, seed]
Determine the extent to which the conditions of equilibrium are verified for an extended object subject to various forces acting in various directions at various points. [lab]

20.01 Relate {incline, alpha, m_1, m_2, weight_1_parallel, weight_1_perpendicular, F_normal, F_frict, weight_2, mu, F_net, a} for mass m_1 on incline whose angle of elevation is alpha, mass m_2 is suspended over pulley at specified end of the incline, mu the appropriate coefficient of friction between mass and incline qa_20

Preliminary:

{m_1, weight_1_parallel, weight_2_perpendicular, alpha}

{m_1, weight_1_parallel, weight_2_perpendicular, mu, F_normal}

20.02 Relate {x_i, theta_i, tau_i, F_i | 1 <= i <= n} U {x_fulcrum, tau_net, F_net} where (x_i, 0) is the point at which the line of action of the force F_i is exerted on a rigid object which includes all the points (x_i, 0), theta_i the angle made by the line of action with the positive x axis, tau_i the torque produced by this force about the point (x_fulcrum, 0), tau_net the net torque, F_net the resultant vector force

Module 3, Assignments 21 - 32

Test #2, cumulative through Module 3, Assigned as part of Assignment 34

Summary of Main Ideas in Module 3

Having completed and reviewed your work in the assignments comprising this Module you should understand these ideas and be able to apply them to solving problems and analyzing real-world situations.

Idea 13: Gravitation is an Inverse-Square Force

The gravitational field has units of acceleration, and in the vicinity of a spherically symmetric planet is inversely proportional to the square of the distance from the center of the planet.

It follows that the gravitational field in the vicinity of such a planet is given by a function g(r) = k / r^2, where k is a proportionality constant and r the distance from the center of the planet.
If we know the value of g at some r beyond the surface of the planet, then we can evaluated the proportionality constant k and obtain a general expression for the field g(r).

The gravitational force between two point masses m1 and m2 is F = G m1 m2 / r^2, where r is the distance between the masses.

Idea 14: For objects in circular orbits, Centripetal Force = Gravitational Force

Centripetal acceleration is given by Fcent = v^2 / r, where Fcent is the centripetal force holding the object in its circular path, v the velocity of the object and r the radius of the circle.

For a satellite in a circular orbit the centripetal force is the net force, which is equal to the gravitational force. For a satellite with mass m in its orbit around a planet of mass M, with M >> m and r = radius of orbit we therefore have m v^2 / r = G M m / r^2, expressing the equality between centripetal force and gravitational force.

This equation is easily solved to obtain an expression for v in terms of orbital radius r.

Idea 15: Change in gravitational PE = Average gravitational force * Displacement

The gravitational potential energy change between two points is equal to the work required to move the object at constant velocity against the gravitational field, from the first point to the second.

The work is equal to the product of the force we must exert against gravity with the displacment parallel to this force.
Since in moving parallel to the gravitational force we change our distance r from the center, we multiply the average gravitation force by the change in r t obtain `dPE = Fave * `dr, where Fave is the average of the (nonlinear) gravitational force on an object and `dr the change in the distance of the object from the center of a planet.
We can approximate Fave by averaging the force at the first distance with the force at the second. However because of the upward concavity of the F vs. r graph this estimate will overestimate the average force.

** To get the accurate change in potential integral we must integrate force with respect to distance from the center. This gives us

** `dPE = integral( G M m / r^2 with respect to r, from r1 to r2), where M is the mass of a planet, m the mass being moved from distance r1 to distance r2 with respect to the center of the planet.

The precise change in potential energy from one distance r = r1 to another distance r = r2 from center is

`dPE = G M m / r1 - G M m / r2

The gravitational potential at distance r from the center of the planet is

gravitational potential = - G M / r.

Gravitational potential is in units of Joules / kg.

The change in potential energy between two points is equal to the change in gravitational potential from the first point to the second, multiplied by the mass of the object being moved.

Idea 16: Between two circular orbits, `dKE = -1/2 * `dPE

For a small mass m in circular orbit around a planet with large mass M, gravitational PE is - G M m / r, while gravitational KE is 1/2 G M m / r.

It follows that to move from a circular orbit at radius r1 to a circular orbit at radius r2 the gravitational PE must increase from - G M m / r1 to -G M m / r2 while KE decreases from 1/2 G M m / r1 to 1/2 G M m / r2.

Gravitational potential - G M / r is negative for all values of r, approaching zero as r approaches infinity. Thus as we move away from a planet we begin with a large negative gravitational potential and as we move further and further away our gravitational potential increases, approaching but never reaching zero.

Gravitational potential energy - G M m / r is also negative for all r, approaching zero as r approaches infinity. If at distance r = R we give the mass m a large enough kinetic energy, so that KE - G M m / r > 0, and if the object moves through empty space so that no energy is dissipated, the work-energy theorem tells us that the increase in PE will never be enough to decrease the KE to zero. That is, the object will never stop.

'escape velocity' is the velocity at which KE - G M m / r = 0
this velocity occurs when KE = G M m / r, so that .5 m v^2 = G M m / r
escape velocity is found by solving this equation for v, obtaining v = sqrt( 2 G M m / r).

Idea 17: Angular Motion is completely analogous to Linear Motion

A one-radian angle any central angle for which the arc distance along a circle is equal to the radius of the circle.

The reasoning for angular motion is identical to that for linear motion, with meters of displacement replaced by radians of angular displacement.

The arc distance corresponding to angular displacement theta is `dsArc = r * theta, where r is the radius of the circle.

Lowercase omega is the Greek letter corresponding to angular velocity, which is the rate of change of angular position and is measured in units of radians per second.

Lowercase alpha is the Greek letter corresponding to angular acceleration, which is the rate of change of angular velocity and is measured in units of radians per second.

The speed of the motion of a point moving around the arc of a circle is equal to the product of the angular velocity of the corresponding radial line and the radius of the circle: v = omega * r.

The acceleration component in the direction of motion of a point moving around the arc of a circle is equal to the product of the angular acceleration of the corresponding radial line and the radius of the circle: vParallel = alpha * r, where vParallel is velocity in (i.e., parallel to) the direction of motion.

Idea 18: Newton's Second Law can be formulated in terms of Angular Motion and Moment of Inertia

The torque exerted by a force is tau = F * r * sin(theta), where tau is the torque exerted by force F applied at distance r from axis of rotation, F making angle theta with moment arm.

In the special case where the force is exerted perpendicular to the moment arm, we have tau = F * r.
Torque in angular dynamics is analogous to force in linear dynamics.

The effect of a net torque on an object depends on the torque and on the difficulty of achieving angular acceleration of the object, in a way analogous to the relationship between mass and net force, where mass is among other things a measure of the difficulty of accelerting an object.

For angular motion the resistance to angular acceleration is measured by the quantity I, called moment of inertia.
For a particle of mass m at distance r from axis of rotation we have I = m r^2.
Once more, moment of inertia is analogous to mass in linear dynamics; just as greater mass requires greater force for a given acceleration, the greater the moment of inertia the greater the torque required to achieve a given angular acceleration.

In general for a real object, I = sum( m r^2), which is the moment of inertia I of a collection of particles of various masses at various distances from axis of rotation.

In general the summation of the m r^2 contributions requires calculus. Among the results we obtain are the following:

For a homogenous circular cylinder with mass M and radius R with axis of rotation through the axis of the cylinder we have I = 1/2 M R^2
For a uniform homogeneous rod with mass M and length L rotating about an axis through the center of the rod and perpendicular to the rod we have I = 1/12 M L^2

Just as Fnet = m a for linear motion, tauNet = I * alpha, where tauNet is the net torque required to achieve angular acceleration alpha on an object with moment of inertia I.

Idea 19: The definitions of Work and KE can be reformulated in terms of Angular Quantites

When we apply torque through an angular displacement we are in fact applying a force through a distance and doing work. Analogous to `dW = F * `ds for linear motion we have `dW = tau * `d`theta, where `dW is the work done by a torque tau acting through angle `d`theta.

Just as work W is equal to the integral of force in direction of motion multiplied by displacement (W = integral ( Fparallel * `ds) we have W = integral ( tau with respect to theta, from theta1 to theta2)

Every particle making up a rotating object has a mass and a velocity and hence a kinetic energy. The kinetic energy of the entire rotating object is the sum of the kinetic energies of all particles making up the object. Particles closer to the axis of rotation have less velocity and hence contribute less to the kinetic energy of the object than particles of the same mass which lie further from the axis of rotation. The moment of inertia I makes it easy to calculate this complicated quantity. We have

KE = .5 * I * omega^2, where KE is the kinetic energy of an object with moment of inertia I rotating about its axis with angular velocity omega. This is analogous to KE = .5 m v^2 for linear motion.

Idea 20: Angular Impulse-Momentum gives rise to a new conservation law.

For linear motion we have impulse `dp = F * `dt, which in a closed system is conserved since objects act on one another with equal and opposite forces. Since rotating objects in contact with one another exert equal and opposite torques on one another they exert equal and opposite angular impulses on one another, where

angular impulse = tau * `dt, where tau is torque and `dt is time interval.

Just as impulse is equal to change in momentum, angular impulse is equal to change in angular momentum, where

angular momentum = I * omega, with I the moment of inertia and omega the angular velocity.

Just as linear momentum is conserved in a closed system, angular momentum is conserved in a closed system.

Angular momentum is not the same as linear momentum. Energy, momentum and angular momentum are all different quantities, each conserved in any closed system.

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21.1

21.2

Conservation of Energy on an Incline

text_21

Elastic and Inelastic Collisions in 1 dimension

#22 - 23

Memorize This: Idea 12

query_21

Objectives:

Observe as accurately as possible the small differences in time required for an object to roll up an incline, and the time to roll back down. [lab]
Analyze energy changes for an object rolling on a ramp. [lab]
Distinguish between vector quantities and scalar quantities. [class notes]

21.01 Relate {h, L, vf_ramp} U {alpha, `ds_y, `ds_x} to test hypothesis that the final velocity vf_ramp of an object rolling a given distance `ds down the ramp from rest depends only on h, where h is the elevation of the higher end of the ramp relative to the lower, L the length of the ramp, alpha the angle of elevation of the ramp conservation of energy on an incline

21.02 Distinguish between vector quantities and scalar quantities introsets, class notes

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22.1

22.2

6.1 – 6.7

Collaborative_labs_II

text_22

Physics I Basic Quantities and Relationships lines 13-24

query_22

Objectives:

Analyze the motion of an object subjected to a constant force not parallel to the object's path. [qa, seed]
Interpret trapezoidal approximation graphs representing various quantities. [introductory problem sets]

22.01 Mass into constant force field: relate {m, v0_x, v0_y, F, theta, `ds_x, `ds_y, vf_x, vf_y} for a mass m which for an interval of motion in a constant force field of magnitude F directed at angle theta, with displacement `ds_x in the x direction and `ds_y in the vertical, and final velocities v0 and vf defined by the components v0_x, v0_y, vf_x and vf_y

22.02 Interpret rise, run, slope, average ‘graph altitude’, accumulated areas, rates of slope change for graphs representing position vs. clock time, velocity vs. clock time, acceleration vs. clock time, force vs. position, force vs. clock time introsets

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23.1

PHeT 8.7 Collision Lab

text_23

Motion in More than 1 Dimension

synopsis: Understand idea 10

query_23

Objectives:

Analyze forces, motion and energy for a chain partially overhanging the edge of a table. [qa]
Analyze drag forces and terminal velocities. [qa, seed]

23.01 Relate {m, L, y_0, v_0, F_net(y), a(y), PE(y), KE(y), v(y), y(t)} for a chain of length L and mass m partially resting on or sliding across and partially overhanging the edge of a table, with y_0 the initial length of the overhang, v_0 the initial velocity of the chain, F_net(y) the net force on the chain as a function of y, y(t) the overhang as a function of t, and a(y), PE(y), KE(y), v(y) the acceleration, velocity, potential energy and kinetic energy functions of y qa_24

Preliminary:

{L, y, F_net(y), PE(y)}

23.02 Relate {v, F_res (v), F, F_net(v), m, v_term} where v is the velocity of a mass m, F_res(v) a resisting force as a function of v, F a constant force, F_net(v) the net force on m when velocity is v, v_term the terminal velocity approached by the object qa_24

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24.1

24.2

Conservation of Momentum

text_24

Motion in 2 dimensions

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Objectives:

Determine the centripetal force of an object moving around a circle at constant velocity. [qa]
Explain why a force is necessary to keep an object moving in a circular path. [qa, seed]
Analyze the projectile motion of two spheres before and after collision. [lab, seed]

24.01 Relate {v, r, a_cent} U {m, F_cent, omega, f, period}, where v is the constant speed of a mass m moving around a circle, r the radius of the circle, a_cent the centripetal acceleration of the object, F_cent the centripetal force on the object, f and omega the frequency and angular frequency of its motion, 'period' the period of its motion around the circle qa_24

24.02 Explain why a force is necessary to keep an object moving in a circular path, even if it doesn't speed up or slow down. qa_24

24.03 Relate {m_1, v_1, m_2, v_2, m_1, v_1’, m_2, v_2’, `dv_1, `dv_2, m_2/m_1} U {`ds_1_x, `ds__2_x, `ds_y } where m_1 and m_2 are the masses of two spheres which collide, moving in a horizontal plane immediately before and immediately after collision before falling a vertical distance `ds_y to the floor under the influence of gravity; `ds_1_x and `ds_2_x are the displacements in the horizontal direction during the fall conservation of momentum experiment

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25.1

text_25

Forces in 2 dimensions; Free Body Diagrams

synopsis: Understand idea 12

query_25

Objectives:

Analyze the internal and external forces for a low-amplitude pendulum. [qa]

25.01 Relate {m, L, T, theta, x, T_x, T_y, F_net, T_string, theta, alpha}, where m is the mass, L the length, x < < L the position with respect to equilibrium, T_x and T_y the components of the string tension, F_net the net force on the mass, T_string the tension of the string, alpha the angle with vertical, theta the angle with horizontal for a simple pendulum qa_25

Preliminary

{m, T_string, T_y}

{T_string, theta, T_x, T_y}

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26.1

text_26

Forces in 2 dimensions; Free Body Diagrams

query_26

Objectives:

Determine the buoyant and net forces on a bobbing cylinder. [qa]

26.01 Relate {A_cs, L, y, y_equil, F_buoy, F_net, rho_cyl, mass_cyl, rho_liquid, a} for a cylinder of length L, uniform density rho_cyl, cross-sectional area A_cs, vertical position y, equilibrium position y_equil, partially submerged in a liquid of uniform density rho_liquid, with F_buoy the buoyant force, F_net the net force, mass_cyl the mass and a the acceleration of the cylinder.

Preliminary:

{A_cs, L, rho_cyl, mass_cyl, weight_cyl}
{y, y_equil, rho_liquid, F_net}
{weight_cyl, F_buoy, F_net}

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27.1

7.1 – 7.7

Motion in a Force Field

PHeT 6.14 Gravity and Orbits

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Circular Motion

#24

synopsis: Understand ideas 13-14

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Objective:

Use proportionality and the known acceleration of gravity at the surface of the Earth to determine the weight of an object at given distance (distance greater than radius of Earth) from the center of the Earth. [qa, class notes]
Use Newton's Law of Universal Gravitation to determine the weight of an object at given distance (distance greater than radius of Earth) from the center of the Earth. [qa, class notes'
Measure the deflection of a moving steel ball as it passes a magnet at different speeds and different distances, and use the results to infer the nature of the magnetic field. [lab]

27.01 Relate {F_grav_satellite, F_grav_planet}, where F_grav_satellite is the force exerted by a planet on a satellite (assumed to be very much smaller and less massive than the planet) and F_grav_planet is the force exerted on the planet by the satellite open qa

27.02 Explain why the acceleration of a small satellite toward a planet is much less than the acceleration of the planet toward the satellite. qa_27, introsets

27.03 Explain the effect on the gravitational force on a satellite of a change in the mass of the planet. qa_27, introsets

27.04 Explain the effect on the gravitational force on a satellite of a change in the mass of the satellite. qa_27, introsets

27.05 Explain the effect on the force on a satellite of a change in the distance between the planet and the satellite. qa_27, introsets

27.06 Relate {F_grav_satellite, F_grav_planet, r, m_satellite, R_planet, g_planet} using proportionalities, where r is the distance between the center of the planet and the satellite, R_planet the radius of the planet and g_planet the acceleration of gravity at its surface . qa_27, introsets

27.07 Relate {F_grav_satellite, F_grav_planet, r, m_satellite, m_planet, G}, where G is the universal gravitational constant qa_27, introsets

27.09 Relate {F_grav, R_Earth, r, g} by proportionality class notes , introsets

27.10 Relate {G, M_planet, r, v} for a satellite in circular orbit about a planet, where v is the orbital velocity at distance r from the center of the planet introsets

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7.8 – 7.12

text_28

Gravitation and Orbital Dynamics

#25 - 27

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Objectives:

Analyze gravitational forces, conditions for circular orbit and orbital energies for a satellite in the vicinity of a planet. [qa, class notes]
Apply the flux picture to determine gravitational forces. [class notes]
Apply the definition of the radian. [class notes]

0.01 Relate {G, M_planet, r, v, PE, KE, m_satellite} for a satellite in circular orbit about a planet, where PE and KE are respectively the gravitational potential energy at distance r and the kinetic energy at orbital velocity v

0.02 …. Univ differential estimates for small orbital changes, derivation of PE formula, integration of F dot `ds along elliptical path, etc.

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8.1 – 8.7

Motion in the Gravitational Field of the Earth

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Collisions, Center of Mass

#28 - 30

Physics I Basic Quantities and Relationships Lines 53-67

synopsis: Understand ideas 13-16

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Objectives:

Apply the definition of radians to radial lines and arcs on a circle of given radius. [qa, introductory problem set, class notes]
Apply the definition of radians to motion of a point on a circle of given radius, and to the definition of angular velocity. [qa, introductory problem set, class notes]
Apply the relationships among angular velocity, radius of circle, speed of motion around the circle and centripetal acceleration. [qa, introductory problem set, class notes]
Make observations to investigate orbital dynamics of a satellite around a planet, using the gravitational field simulation, and reconcile observations with theory. [lab]

29.01 Relate {r, `dTheta, `ds_arc} for a two positions on a circle of radius r, where `dTheta is the angle in radians between radial lines intersecting the points and `ds_arc the distance between the points as measured along the arc of the circle qa_29, introsets

29.02 Relate {r, omega, v_arc} for a point moving along the arc of a circle of radius r with angular velocity omega and (tangential) speed v_arc along the arc of the circle qa_29, introsets

29.03 Relate {r, omega, v, a_cent} where a_cent is the centripetal acceleration of the point qa_29, introsets

29.04 Relate {A, omega, theta_0, t, r_x, r_y, v_x, v_y, a_x, a_y} where theta_0 is the angular position at t = 0 of a point moving with angular velocity omega around the arc of a circle of radius A, t a clock time, r_x and r_y are the x and y coordinates of the position of the point, v_x and v_y the components of the velocity vector, a_x and a_y the components of the centripetal acceleration vector qa_29, introsets???

29.05 Describe the effect of increasing or decreasing speed on the shape of a circular orbit. Motion in the gravitational field of the Earth

29.06 Describe the effect of a change in velocity in a direction perpendicular to a circular orbit. Motion in the gravitational field of the Earth

29.07 Describe the way velocity and distance from the planet change in an elliptical orbit. Motion in the gravitational field of the Earth

29.08 Describe how consistent increments in the velocity of a ‘shot’ perpendicular to the surface of the Earth affect how far each ‘shot’ will ‘rise’ relative to the preceding. Motion in the gravitational field of the Earth

29.09 Describe how to move from one circular orbit to another. Motion in the gravitational field of the Earth

29.10 Describe the effects of initial speed, direction and position on the shape of an orbit: {v0, v_circ, theta_0, x_0, y_0, shape of orbit} where v0 is the initial speed, v_circ the orbital speed for a circular orbit through the initial point, theta_0 the angle of the initial velocity with respect to the x axis, x_0 and y_0 the coordinates of the initial point in the plane of the orbit, and 'shape of orbit' is its eccentricity class notes? , introsets

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8.8 – 8.18

PHeT:

9.6 Torque
10.7 Ladybug Revolution

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Rotational Motion; Torque

synopsis: Understand ideas 17-18

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Objectives:

Analyze uniformly accelerated angular motion on an interval. [qa]
Incorporate moment of inertia and net torque into the analysis of uniforly accelerated angular motion. [introductory problem sets]

30.01 Relate {omega_0, omega_f, alpha, `dt, `dTheta, `dOmega, omega_Ave} assuming uniform rotational acceleration during an interval, where omega is angular velocity, theta is angular position, `dt is time duration of the interval and alpha is angular acceleration qa_30, introsets

30.02 Relate {I, tau_net, alpha} U {omega_0, omega_f, `dt, `dTheta, `dOmega, omega_Ave} where I is the moment of inertia of an object rotating about an axis or a point and tau_net the net torque exerted on the object (torque with respect to point or axis of rotation) qa_30, introsets, class notes?

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View Experiment 29, Angular Velocity and Velocity at Given Radius (Physics 121 may omit this experiment)

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Moment of Inertia, Rotational Dynamics, Angular Momentum

#31 - 33

Physics I Basic Quantities and Relationships Lines 53-67

synopsis: Understand Ideas 19-20

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Objectives:

Know and apply the formulas for moments of inertia of specified disks, hoops, spheres and rods. [class notes, qa, text]
Analyze the dynamics of a rotating object subject to the torque of a descending mass, including analysis of motion and energy. [class notes, qa, text]
Apply the conditions of equilibrium to extended objects. [class notes, text]

31.01 Relate {I, disk, hoop, sphere, rod about end, rod about center, M, R, L} for moments of inertia of disks, hoops, spheres and rods qa_31, introset

31.02 Relate {r_axel, m_descending, I, `ds_y, `dPE, `dKE, omega_0, omega_f} for a wheel with moment of inertia I with descending mass on string wrapped about a light disk, wheel and disk rotating about a common axis, on an interval during which vertical position changes by `ds_y, gravitational potential energy by `dPE, total kinetic energy by `dKE, abd angular velocity changes from omega_0 to omega_f.

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View Experiment 30, Conversion of Gravitational or Elastic Potential Energy to Angular Kinetic Energy

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Moment of Inertia, Rotational Dynamics, Angular Momentum

Outline of Angular Quantities and their Relationships

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Objectives:

Find the moment of inertia of a composite object and apply to the analysis of rotational kinematics and dynamics. [qa]
Apply conservation of energy to systems whose motion includes angular kinetic energy. [qa, lab]

32.01 Relate {`dPE, I, `dKE, omega_0, omega_f} where I is the moment of inertia of a combination of disks and/or spheres and/or rods and/or hoops and/or a specified set of discrete masses and positions, omega_0 and omega_f the initial and final angular velocities, `dKE and `dPE the changes in potential and kinetic energy during an interval experiment 30

Module 4, Assignments 33 - 40

Final exam cumulative through Module 4, with extra emphasis on Module 4

Summary of Main Ideas in Module 3

Having completed and reviewed your work in the assignments comprising this Module you should understand these ideas and be able to apply them to solving problems and analyzing real-world situations.

Idea 21: Simple Harmonic Motion results from a Linear Restoring Force

When the net force Fnet acting on a mass m is proportional to displacement of the object from some equilibrium position x we can write Fnet = - k x. The proportionality constant k is called the force constant.

Since Fnet = m a we have m a = - k x, or a = -k/m * x.

** Since a(t) = x '' (t) it follows that x(t) = B sin( omega * t) + C cos(omega * t) = A cos(omega * t + theta0), where omega = sqrt(k/m) and theta0 can be any real number.

When this proportionality exists between Fnet and x, an object displaced from equilibrium and released will oscillate about the equilibrium position with angular frequency omega = sqrt( k / m ).
The motion of such an object is called Simple Harmonic Motion.
The detailed motion can be described by the equation x(t) = A cos( omega * t ), where A is the maximum displacement from equilibrium. A is called the amplitude of the motion. This model assumes that the object reaches its maximum positive displacement from equilibrium at clock time t = 0.
If the model x(t) = A cos( omega * t ) describes the motion then the velocity is described by the equation v(t) = -omega * A * sin(omega * t).
** the velocity function v(t) is the derivative v(t) = x ' (t) = dx/dt of the position function x(t)
If the model x(t) = A cos( omega * t ) describes the motion then the acceleration is described by the equation a(t) = -omega^2 * A * cos( omega * t ).

Note that this acceleration is not uniform. Simple harmonic motion cannot be analyzed using the equations or the reasoning processes we use with uniformly accelerated motion.

** the acceleration function a(t) is the derivative a(t) = v ' (t) = da/dt of the velocity function v(t), the second derivative a(t) = x '' (t) of the position function

* If the position is not at the maximum positive displacement when t = 0 then the model x(t) = A cos( omega * t ) is not appropriate. However a slight modification can give the position function corresponding to any specific initial condition:

* If the initial position is not at the maximum positive displacement, choosing an appropriate theta0 for the general equation of motion x(t) = A cos( omega * t + theta0 ).

* Another possible form of the general equation is x(t) = A sin( omega * t + theta0 ) .

*Using either form theta0 must be chosen to satisfy specified conditions on the motion.

* The position function is x(t) = A sin( omega * t + theta0) gives us velocity function v(t) = -omega * A * sin(omega * t + theta0).

** the velocity function is the derivative of the position function

* The position function is x(t) = A sin( omega * t + theta0) gives us velocity function a(t) = -omega^2 * A * cos( omega * t + theta0 ).

** the acceleration function is the derivative of the velocity function, the second derivative of the position function

* 21.10. Other possible position functions include x(t) = A sin(omega * t) or y(t) = A sin(omega * t) or x(t) = A sin(omega * t + theta0) or y(t) = A sin(omega * t + theta0).

Idea 22: Simple Harmonic Motion can be modeled by the projection of a point moving about a Reference Circle

Simple harmonic motion can be modeled as the projection onto the x axis (or onto the y axis, or indeed onto any axis thru the center of the circle) of motion at constant angular velocity around a reference circle whose radius is equal to the amplitude of the motion.

The forms x(t) = A cos(omega * t), x(t) = A cos(omega * t + theta0) and y(t) = A sin(omega * t + theta0), as well as others, can be obtained from the circular model:

If motion around the reference circle at angular velocity omega starts at the positive x axis when t = 0 then x(t) = A cos(omega * t) is the x coordinate of that point at clock time t.
If motion around the reference circle at angular velocity omega starts at angular position theta0 when t = 0 then x(t) = A cos(omega * t + theta0) is the x coordinate that point at clock time t.
If motion around the reference circle at angular velocity omega starts at the positive x axis when t = 0 then y(t) = A sin(omega * t) is the x coordinate of that point at clock time t.

Idea 23: Velocity and Acceleration in SHM follow the Reference Circle Model

The formulas given earlier for v(t) and a(t) can be found directly from the circular model:

The reference-circle point, which moves on a circle of radius A with angular velocity omega has speed v = omega * A and is directed tangent to the circle, so that the velocity of the reference-circle point is v = omega * A at a right angle to the radian line (i.e., the line from the center of the circle to the reference point).
The centripetal acceleration of the reference-circle point is a = v^2 / A = omega^2 * A and is directed toward the center of the circle.
If the reference-circle point starts at the positive x axis when t = 0 then at clock time t the radial line makes angle omega * t with the positive x axis, so that at clock time t the velocity vector is at a right angle to omega * t. It follows that the x component of the velocity of the reference point is vx(t) = -omega * A * sin(omega * t).
If the reference-circle point which starts at the positive x axis when t = 0 then the direction of the centripetal acceleration is opposite to the direction omega * t of the radial line. It follows that the x component of the centripetal acceleration is ax(t) = -omega^2 * A * cos(omega * t).

Note that reference-circle velocity is parallel to the x axis when the reference circle point lies on the y axis. At these points the reference-circle angle omega * t is at a right angle with the x axis so that | sin(omega * t) | = 1 and | vx(t) | = omega * A.

Thus when the reference circle angle crosses the y axis, the x velocity matches the speed of the point on the reference circle.

Note also that reference-circle centripetal acceleration is parallel to the x axis when the reference circle point lies on the x axis. At these points the reference-circle angle omega * t is parallel to the x axis so that | cos(omega * t) | = 1 and | ax(t) | = omega^2 * A.

Thus when the reference circle angle crosses the x axis, the x acceleration matches the centripetal acceleration of the point on the reference circle.

Idea 24: Energy Relationships in SHM are consistent with the Reference Circle Model

The PE relative to the equilibrium position of a simple harmonic oscillator at displacement x from equilibrium is the work done against conservative forces in order to displace the oscillator from equilibrium to that position.

The restoring force Fnet = - k x is conservative. The work done against this conservative force will be the negative of the work done by the conservative force.
The work done by Fnet between equilibrium and position x is equal to Fave * `ds, where Fave is the average value of Fnet between the two positions and `ds is the displacement.
Since Fnet changes linearly from zero at equiibrium to -k x at position x its average value is Fave = (0 + (-kx) ) / 2 = -k x / 2. This simple average works only because Fnet is linear with respect to x.
Displacement from equilibrium to position x is `ds = x.
Thus the work done by the restoring force is Fave * `ds = -kx / 2 * x = - k x^2 / 2.
The work done against the restoring force is `dPE = - (-k x^2 / 2) = k x^2 / 2.

Thus the PE of the oscillator at position x, relative to equilibrium, is PE = k x^2 / 2.

The PE of the oscillator is therefore maximized at the extreme positions x = A and x = -A.
At maximum amplitude we have PE = .5 k A^2.
At maximum amplitude the velocity of the oscillator is zero so its KE is zero.
The total mechanical energy of the oscillator at maximum amplitude is therefore just its PE
Since the net force Fnet = - k x on the oscillator is conservative there are no dissipative forces and conservation of energy tells us that `dPE + `dKE = 0, so that PE + KE must be constant. It follows that for a simple harmonic oscillator
total mechanical energy = PE + KE = 1/2 k A^2.

Since at position x we have PE = 1/2 k x^2, conservation of energy tells us that

1/2 k x^2 + KE = 1/2 k A^2 so that
KE = 1/2 k A^2 - 1/2 k x^2.

At position x = 0 we have PE = 0 so that velocity is maximized at this point and

KE at equilibrium = 1/2 k A^2
1/2 m vMax^2 = 1/2 k A^2 so that
vMax = sqrt( k A^2 / m). Since omega = sqrt(k/m) we have
vMax = omega * A.

Recall that from the circular model we have already concluded that vMax = omega * A. Thus the energy picture is consistent with the circular model, which itself is consistent with the model obtained using Newton's Second Law and calculus.

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9.1 – 9.5

PHeT 16.15 Pendulum Lab

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Statics

Physics I Practice Tests Assignment 3 (Test 2)

synopsis: Understand Ideas 19-20

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· Go to Tests and investigate Test #1. For Physics 121 this test will cover Introductory Problem Sets 5, 6, 7 and 8 plus experiments between Assignment 20 and Assignment 32. For Physics 201, 231 and 241 the test will cover material starting with Assignment 1, but will concentrate on material in Assignments 20 - 32.

· Recommended but not required: Run the file Physics I Practice Tests Assignment 3 (Test 2) for practice on the upcoming Test 2.

Objectives:

Calculate and apply the concept of angular momentum. [qa]
Show that conservation of angular momentum follows from Newton's Third Law [qa]
Find and apply the moment of inertia of an ideal rigid object consisting of specified discrete point masses. [qa]
Use the circular model to find the position function of a simple harmonic oscillator. [introductory problem sets]

33.01 Relate {I, omega, KE_angular, L} where I is moment of inertia and omega angular velocity about an axis, KE_angular is angular KE, L is angular momentum qa_33

33.02 Relate {I_1, I_2, tau_net, `dt, `dL_1, `dL_2, `dOmega_1, `dOmega_2} where subscripts _1 and _2 refer to two rigid objects with I, L, tau and omega relative to a fixed point or axis . tau_net is the magnitude of the torque exerted by one object on the other.

33.02 Relate {omega, r_i, m_i, KE_rot_i | 1 <= i <= n} U {KE_rot, L} where omega is the angular velocity of a rigid system consisting of n discrete masses, r_i and m_i the the ith mass and its distance from the axis of rotation, KE_rot_i the rotational KE of this mass, KE_rot the rotational KE and L the angular momentum of the entire system qa_33

33.03 Relate {k, m, A, omega, t, x, y, theta_0}, where k is the restoring force constant, m the mass, A the amplitude, omega the angular frequency of the reference point, x and y the components of the position vector on the reference circle, t the clock time, theta_0 the initial angular position on the reference circle for a low-amplitude pendulum or general simple harmonic oscillator

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9.6 – 9.11

(text_34)

#34 - 35

synopsis: Understand Ideas 22-24

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Complete Test #2

Objectives:

Apply the reference circle model for the dynamics of simple harmonic motion. [qa, introductory problem set]

34.01 Relate {k, m, A, omega, t, x, y, init cond, vAve on short interval near equil, speed of reference point} for a harmonic oscillator where 'init cond' stands for the initial condition of the oscillator (restate problem) , qa_34

34.02 Relate {m, init cond, k, vel on ref circle, a_Cent, a_max}, qa_34

34.03 Relate {period, mass, force constant, frequency} for a simple harmonic oscillator, qa_34

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9.12 – 9.17

(text_35)

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Objectives:

Apply the reference circle model to analysis of the kinetic, potential and total energy of a simple harmonic oscillator. [qa, introductory problem sets]

35.01 Relate {k, x_1, x_2, F_ave, F(x_1), F(x_2), `dW, `dKE} for a simple harmonic oscillator on an interval between positions x_1 and x_2 qa_34

35.02 Relate {k, m, A, x, PE, KE, PE_max, KE_max, total energy, omega, f, period} for a simple harmonic oscillator with amplitude A, where PE and KE are the potential and kinetic energies at position x

34.03 Relate {L, m, k, omega, f} for a pendulum, where f is the frequency, qa_34

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