DRAFT

Class 090923

<h3>The average rate of change is (change in voltage) / (change in clock time). This has nothing to do with dividing by 2.

If you divide the difference of two values of a quantity by two, you get half the difference, which isn't involved with the rate-of-change definition.

The average rate is (change in flux) / (change in clock time) = ( .816 T m ^ 2)/( .06755 sec) = 24.15 T m ^ 2 / sec = 24.15 Volts.</h3>

The following conventions will allow your instructor to quickly locate your answers and separate them from the rest of any submitted document, which will significantly increase the quality of the instructor's feedback to you and to other students.

When answering these questions, give your answer to a question before the &&&&.  This is different than my previous request to place your answer after the &&&&.

When doing qa's, place your confidence ratings and self-assessment ratings on the same line as the prompt.

If you don't follow these guidelines you may well be asked to edit your document to make the changes before I can respond to it.

Thanks.

Trapezoidal Graph of Rubber Band Calibration Data

When you calibrated your rubber band chain you observed the position of the end of the chain with various numbers of dominoes. 

Plot the information on a graph of y = number of dominoes vs. x = position of end.  Your graph will consist of five points.

Connect your five points with line segments.  There will be four line segments, each with a slope you calculated previously.

From each point sketch the 'vertical' line segment from that point to the horizontal axis.  You will sketch five line segments, each representing the number of dominoes suspended in the corresponding trial.  Having sketched these segments, you will have constructed a series of four trapezoids.

Find the area of each trapezoid.

Label the slope of each trapezoid by placing it in a rectangular 'box' just above the 'slope segment' at the top of the trapezoid.

Label the area of each trapezoid writing it inside the trapezoid and circling it.

What are the four slopes?

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What are the four areas?

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What does each slope represent (explain what the rise represents, what the run represents, and what the slope therefore represents; be sure to include units with each quantity):

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What does each area represent (explain what the altitude of the equal-area rectangle represents, what the base represents, and what the area therefore represents).

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Correct accelerations for lab situations:

In a number of lab situations acceleration is assumed constant, and we observe the time needed for an object to undergo a known change in position, either starting or ending at rest.

To solve the motion in such a situation we the following procedure will always work efficiently:

• Apply the definition of average velocity.
• Sketch the v vs. t trapezoid.
• Use the trapezoid to determine the initial and final velocity of the object on the interval.
• Apply the definition of average acceleration.

One thing that many students do is divide average velocity by time interval (vAve / `dt).  This is not a calculation that comes up when following the above procedure, and it isn't a calculation that tells us anything important about the motion.  Be sure you understand why this calculation doesn't happen in the given procedure.

Another common error is for students to double the average velocity to get the final velocity.  For the case where initial velocity is zero and acceleration is constant, this is actually done.  However this is a result of the fact that the v vs. t trapezoid is in this case a triangle, and is not something we generally do.  You shouldn't even think about doubling the average velocity, and certainly shouldn't get into the habit of doing so.  Draw the trapezoid and do as the drawing dictates.

Being very sure you analyze the motion correctly, please report the following.  If you've done some or all of these correctly, you can insert a copy (or copies) from your previous document(s):

The data and accelerations of the three trials for the Atwood machine:

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The two slopes obtained when you graph acceleration vs. number of added rubber bands for the Atwood machine:

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The data and accelerations for the three trials (domino flat, on long side, on short side) for the ball-down-ramp experiment:

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The two 'graph slopes' obtained when you graph acceleration vs. ramp slope:

2-ramp expt

Newton’s Second Law

Here are a few fairly obvious statements. 

·         It takes more force to accelerate lots of mass than just a little mass.

·         The more acceleration you want the more force you need.

·         More force implies greater acceleration.

All these statements are a bit imprecise, requiring a bit of interpretation.  More precise statements require more words, with a potential for confusion.  So we’re going to work from the imprecise statements that first form our ideas, to the precise statements on which we base the actual science.

For example more precise and accurate versions of the above statements might look like the following:

To give two different masses the same acceleration, the greater mass requires the greater force.

To give a certain mass a greater acceleration, you need to apply a greater force.

If you apply the same force to two different masses, the lesser mass will have the greater acceleration.

Not really so.  You can apply all sorts of force to a merry-go-round and it never goes anywhere, it just rotates, with the parts near the rim moving faster than the parts near the center.

So we want to talk about situations that don’t involve rotation.

Particles don’t rotate.  They only translate.

These are summarized

The net force exerted on a particle is the product of its mass and its acceleration:

F_net = m a.

The net force on a particle is what you get if you combine all the force acting on it.

A couple of important consequences:

If its acceleration is zero then the net force on it is zero.

If it’s sitting still then its acceleration is zero.  If it’s moving at constant speed in a constant direction then its acceleration is zero.  Otherwise its acceleration isn’t zero and the net force on it is not zero.

Substituting a = F_net / m into the second and fourth equations of motion:

If we solve Newton’s Second Law for acceleration we get

a = F_net / m

If substitute F_net / m for a in the second equation of uniformly accelerated motion, vf = v0 + a `dt, what equation do we get?

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We can rearrange your equation to get

·         F_net * `dt = m vf – m v0

Show the steps needed to get from your previous answer to this form:

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If substitute F_net / m for a in the fourth equation of uniformly accelerated motion, vf^2 = v0^2 + 2 a `ds, what equation do we get?

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We can rearrange your equation to get

·         F_net * `ds = ½ m vf^2 – ½ m v0^2.

Show the steps needed to get from your previous answer to this form:

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Force vectors

The three arrows you drew to depict the forces exerted by rubber bands are called force vectors. 

If you put the arrows end-to-end they should ideally form a ‘closed’ triangle (i.e., the last arrow should end where the first one began).  This would indicate that when the forces are combined, they add up to zero.

If two of the rubber bands exert forces in the x and y directions, one force in the x direction and the other in the y direction, then the third force needs to be in a direction that balances both the x and the y forces.  …

trap gph for your rb chain

how much stretch for extra domino at various legnths; 10% stretch; etc

divide into 3-cm trapezoids

divide into .5-domino trapezoids

F = m a (lesson of Atwood)

vectors; rb chain opposes two other rbs (or use rb chains from three different groups ...)

sketch force vectors

sketch force vectors for ball on incline

avoid superstition

demo impart energy to rotating strap; measure period to estimate KE_rot (identify translation; is there more energy in rotation or in translation)

can you determine which car has less friction without anyone or anything changing sides of the room?

algebra problem:  substitute F / m for a in the second equation then solve for m (then for mv?)

substitute F / m for a in 4th equation then eliminate denominators

what's that in front of your face?  (2 ramps)

ave roc of v wrt t is not vAve / `dt.  Never was.  Never will be.  Ain't so.  Contradicts the definition.  Pure superstition.  Balderdash.  Not complete nonsense but yer thinkin went haywire.

6 4

inFrontaYerFace qa

bounce frequency apparently not uniform; consider reasons why (if force vs. length linear and rb never collapses, then should be uniform; conclusion if not uniform then not linear)

what do you get when you divide change in vel by change in clock time, ave. vel by change in clock time, which is related to motion and why, which isn’t and why

Acceleration vs. ramp slope

If you missed class today you can easily complete this experiment by staying an extra 5 minutes or so next time.  It's a familiar situation (ramp supported by a domino lying flat, then on its long side, then its short side).  You will be asked to calculate the acceleration of the ball and the slope of the ramp, create a three-point graph, and calculate two graph slopes.

• By lining up dominoes in their proper orientation, as demonstrated in class, determine the slope of your ramp when supported by a domino lying flat, then lying along its longer edge, then along its shorter edge.
• Using your pendulum in the manner you deem most accurate, take observations necessary to find the acceleration of the ball on each ramp.  Try to determine the time down the ramp as accurately as possible.

Give your data and an explanation of what they mean:

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Graph acceleration vs. ramp slope and give the three graph points you get as a result of your analysis of the data.

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Find the slope between the first and second point on your graph.

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Find the slope between the second and third point on your graph.

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What do you think is the uncertainty in your time measurements?

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What do you think is the percent uncertainty in each of your time measurements?

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What do you think is the percent uncertainty in each of your calculated accelerations?

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What do you think is the percent uncertainty in each of your slopes?

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When you analyze your data for this experiment:

For what object are you calculating the acceleration?

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What events define the beginning and the end of each time interval you are measuring?

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Where is the object at the beginning of the interval and where is it at the end of the interval?

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What is the displacement between those positions?

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Does the displacement depend in any way on the length of the pendulum?

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What information do we get from the length of the pendulum?

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Interpreting Trapezoids

A trapezoidal approximation graph has 'altitudes' 70, 90, 120 and 130 corresponding the 'horizontal' coordinates 120, 150, 190, 220.

We sketch the graph and complete the three corresponding trapezoids.  We calculate their slopes, then construct the equal-area rectangles and use them to calculate the areas of the trapezoids.

We easily obtain the following information:

Now we interpret this, assuming that the graph represents velocity in cm/s vs. clock time in s.

The vertical quantity is velocity in cm/s, and the horizontal quantity is clock time in s.

The rise of a trapezoid therefore represents the corresponding change in velocity, designated `dv, in cm/s.

The run of a trapezoid represents the corresponding change in clock time, designated `dt, in seconds.

The slope is rise / run = `dv / `dt, in (cm/s)/s = cm/s^2.  This is change in velocity / change in clock time, i.e., average rate of change of velocity with respect to clock time, which is the definition of acceleration.

The average altitude is the average of the two altitudes, which represent velocities.  The average altitude therefore represents the approximate average velocity on the interval.  We can designate this as vAve.

The area is the product of the average altitude and the run, which represents the product of the average velocity vAve and the change in clock time `dt.  The average velocity is the average rate of change of position with respect to clock time, which by definition is change in position / change in clock time, or `ds/`dt.  So when we multiply vAve by `dt we are multiplying the approximate value of `ds / `dt by `dt, and the result is the approximate value of `ds, the change in position.  Thus the area of the trapezoid represents the approximate change in position during the corresponding interval.  In this case the units are cm/s * s = cm.

We conclude that the object whose velocity is represented by the graph has average accelerations 2/3 cm/s^2, 4/3 cm/s^2 and 1/3 cm/s^2 on the three respective intervals, and travels approximate distances of 2400 cm, 4200 cm and 3750 cm.

The instructor will again run an Atwood machine experiment, with you doing the timing.  Write down your data carefully. 

We can also interpret this data assuming a graph of force in pounds vs. position in feet.  To visualize a possible situation, think of the force exerted by a bungee cord as the position of its free end changes (thereby stretching it):

The rise of a trapezoid will represent the change in the force `dF in lbs.

The run will represent the change in position `ds, in ft.

The slope will therefore represent `dF / `ds, i.e., change in force / change in position.  By definition of rate this is the average rate of change of force with respect to position for the interval represented by the trapezoid.  Its units will be lbs / ft.  This tells you how many pounds of force you can expect for each foot you stretch the bungee cord.

The average altitude of a trapezoid will represent the approximate average force F_ave on the corresponding interval, in lbs. 

So the area of a trapezoid represents the product of the average force and the change in position, designated F_ave * `ds, in units of ft * lbs.  (With the bungee-cord interpretation, if the position is changed in the direction which stretches the cord, this is the work done to stretch the cord.  If the cord is an 'ideal' elastic object (which is not completely so), this also represents the potential energy stored in the cord.)

qa assignments

There is a series of over 30 q_a_ exercises, plus an additional set of qa's designed to address areas of difficulty indicated by submitted work.  Copy the address into the Address box of your browser:

http://vhcc2.vhcc.edu/ph1fall9/frames_pages/qa_grid.htm

It is recommended that you bookmark this page.

To help you prepare for the upcoming Major Quiz, you should plan to run through qa's #2 - 8 during the upcoming week.  If you do one a day you'll be done by the first of next week.  The first few should go pretty quickly, and if you've mastered the content to this point of the course none of them should take a lot of time.  In any case you'll find that they are worth the time you put into them.

There are at least two ways to run through these documents:

• If you have mastered all or most of the content of a document (as will be the case for most of the class on the earlier documents) then you can simply read each problem, think through in your head how you would solve it, then read the given solution and see if there's anything you didn't think of. 

• If you thought of everything, you can just move on to the next question.  If you don't insert an answer, confidence rating, self-critique and self-critique rating, it will be taken as an indication that you fully understand the problem and its solution and would be able to use the corresponding ideas to solve similar problems.
• If there's something you didn't think of, then if you are very sure you understand the details you didn't think of, you can move on, but it's recommended you jot down some notes in your notebook.  If you're not sure of the details, then you should insert a detailed self-critique in which you demonstrate what you do and do not know, ask questions, etc., according to the instructions for the self-critique process.
• If you're not completely sure, you should stop and insert a detailed self-critique.

• If you have not mastered the content, or if you discover that at some point you aren't sure, then you should at that point begin inserting your solutions, confidence assessments and self-critiques.

Homework:

Your label for this assignment: 

ic_class_090921

Copy and paste this label into the form.

Report your results from today's class using the Submit Work Form.  Answer the questions posed above.

Note also that you will receive a subsequent document with some alternative materials, and that you will be asked to complete a short portion of that document.