Class 091019

address 121 vs. 201 (focus on labs and problem sets)

atwood from last time; friction; move into analyzing frictional forces on incline

what would you feel if you were riding the Atwood (elevator problem)

text ch 2, 4, 6, 7; find problems ...

Set up an Atwood machine using dominoes and clips.

• Choose a positive direction and start adding paperclips to the appropriate side until you get positive acceleration.  Measure the minimum possible positive acceleration, and note the number of clips that have been added.  Then add two more clips and repeat. 
• Remove all added clips and start adding clips to the negative side until you get the smallest possible negative acceleration, which you should measure.  Note the number of clips that have been added.  Then add two more clips to the negative side and repeat.
• Keep track of every 'move', including every mass added to either side of the system.

`q001.  Report your data for the Atwood machine.  Report all the raw data you will use in the analysis, in which you will figure out the four accelerations and the total effect of the gravitational force in accelerating each system.

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`q002.  Figure out the acceleration for each system, and report below, with a brief synopsis of how you proceeded from your raw data to your conclusions.

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`q003.  Find the total effect of the gravitational force on each system, as a multiple of the weight of a paper clip.  Paper clips added on one side exert positive force, clips added on the other side exert negative force on the system.

For example if you have 9 clips on one side and 5 clips on the other, the net gravitational force is equal to the weight of 4 clips.

Give a table of acceleration vs. total gravitational force.

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`q004.  Your table should include a fairly large 'gap' in the total gravitational force, where the acceleration is zero.  How wide is that 'gap'?  What does that 'gap' mean?

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Suspend some number of dominoes, between 3 and 7, from your rubber band chain and measure its length.  Don't let anyone know how many dominoes you suspended.  Measure the length of the chain.  Measure once more with the chain supporting a single domino.  Then proceed as follows:

• Get together with 2 other people in the class and hook all three chains to the same paper clip. 
• Choose one person to provide the equilibrant force. 
• The other two will pull their chains back until they are at their previously measured lengths, keeping them at a right angle, while the third provides the equilibrant. 
• When the system is in this state, the person providing the equilibrant will measure the length of his or her chain, and data will be taken to determine the angles between the first two rubber bands and the equilibrant. 
• The two people whose rubber bands are at right angles will share the numbers of dominoes that resulted in the given length. 
• The lengths of all three rubber bands for this trial will be shared.

Then the experiment will be repeated twice more, so that it has been repeated once with each person providing the equilibrant.

If one or more groups of four people are required, adjust the experiment so that each individual provides the equilibrant force one time.

`q005.  You will find the magnitude of the resultant of the two forces exerted at right angles, the angle made by the equilibrant with these two forces, and the estimated force exerted by the equilibrant, based a two-point graph of number of dominoes supported vs. chain length.

Report the raw data you will use to find these quantities.

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`q006.  Find the magnitudes of the two forces, the angle of the equilibrant and the force exerted by the equilibrant, as estimated from your data.  Briefly indicate how you found these quantities.

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`q007.  Sketch the first two forces on a set of coordinate axes, and find the magnitude and angle of their resultant.  Give your results and your explanation of the calculation below.

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`q008.  Sketch the equilibrant on the same set of coordinate axes, and find its x and y components.

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`q009.  Compare the observed equilibrant with the resultant:

• To what extent is the equilibrant obtained from your data equal and opposite to the resultant of the other two forces? 
• By how much does its magnitude differ from that of the resultant?   
• By how much does its angle differ from that of the resultant? 
• By how much do its components differ from those of the resultant?

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`q010.  A ball rolls down an incline of length 30 cm with slope 0.1, and falls to the floor 1.05 meters below.

A slope of 0.1 can be associated with a rise of 0.1 and a run of 1.0.  A vector with a rise of 0.1 and a run of 1.0 has a y component of 0.1 and an x component of 1.0, so it makes angle

theta = arcTan(y comp / x comp) = arcTan (0.1 / 1.0) = 5.7 degrees

with the positive x axis, as measured in the counterclockwise direction.

A ramp slope of 0.1 can also be associated with a vector whose rise is either +0.1 or -0.1, and whose run is either +1.0 or -1.0.

• What possible angles might the vector therefore make with the positive x axis?

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`q011.  The ball comes off the bottom of the incline with its velocity in the direction of the ramp.  What therefore are the possible angles, in the coordinate plane, of the velocity vector with a horizontal x axis?

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`q012.  If the ball requires 1.1 seconds to roll the length of the ramp, what is the magnitude of its velocity vector as it comes off the end of the ramp?

What therefore are the x and y components of its velocity vector at this instant?

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`q013.  The initial vertical velocity of the ball is equal to the y component of its velocity as it comes off the ramp.  Its acceleration in the vertical direction is that of gravity. 

• Use this information and the distance of the fall to determine the time required for the ball to fall to the floor.

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`q014.  The ball's horizontal velocity is constant, and equal to the horizontal component of its velocity as it comes off the ramp.  Using this with the time interval determined previously, how far does it travel in the horizontal direction as it falls?

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`q015.  Assuming the ball to be a sphere of density 8 grams / cm^3 and diameter 2.5 cm, find its kinetic energy at each of the following instants:

• when it is released
• when it leaves the end of the ramp
• when it reaches the floor

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`q016.  Continuing the preceding, assuming that the ball descends 3 cm on the ramp:

• What is its PE change while on the ramp, and what is its PE change while falling to the floor?
• How much work is therefore done by the frictional force between the ball and the ramp?

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`q017.  Answer for your rubber band, for the number of dominoes you chose to suspend from it:

Assuming that each domino weighs .2 Newton, and making the approximation that the force vs. length curve for the rubber band is linear, what is the average force required to stretch the rubber band from its 1-domino length to the length corresponding to your chosen number of dominoes?

How much work is therefore required to stretch it between these two lengths?

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... angles, forces for 3-domino ramp; for 24 cm pendulum pulled back 4 cm; for initial velocity of ball off end of 4-domino incline; for a 3- and a 5-domino rubber band opposed to another of given (linear) calibration graph

... energy stored in rubber bands

... const-vel ramps

... do we have a rich experiential base yet?  can we create one with D?  pictures and story construction, bringing multiple skills to bear on a complex problem

on experiments: ability to get data quickly touted as excuse for excessive use of computer-interfaced instruments; another option is to use really simple instruments, which in reasonable combination with interfaces etc. are more than sufficient to the task of learning physics and experimental technique

do expt with 3 rb, then sketch and estimate three vectors corresponding to claimed forces (see how long with your chosen # of dominoes (3-7), 'war' one against 2 others at 90 deg to find equilibrant, analyze

3 dominoes instead of 1 on 2-ramp expt; does apparent coeff of rolling friction change and if so by how much; or accel on const-vel ramp .... use screw mechanism

labs:  establish basic instruments and their properties:

• pendulum
• incline
• horiz and near-horiz projectile
• incline-projectile
• rubber bands and chains
• atwood machine
• rotating strap
• magnet and steel ball ...

revisit F vs. x graphs for rbs

Atwood graph for neg accel; effect of friction on graph.  Either counterbalance both ways, graph a vs. wt_net, discover friction; or accel in neg direction etc..

`dt for intercepting ball at perpendicular, at angles (collision cs changes with angle); why not just let the thing go across the floor and check out the reaction?  optimize angle of ramps (head-on not good because of uncertainties in path, perpendicular probably not good because of lesser cross-section), steepness of 'missile' ramp (to some extent steeper means less time delay between release and result; however too fast reduces chance of target hitting 'missile' as opposed to reverse ... )

revisit v vs. t graph for ball down ramp

resolve forces on incline and for pendulum

init velocity of ball off inclined ramp (how much error to assume horiz velocity is uniform?)

is that 90% cm horiz range really possible for 24 cm pendulum?

some might work better from equations ... provide later in this document ... interpret this equation in terms of our lab experiences

qa's 15-18 (impulse-mom, vectors; 19 is on vector quantities)

20 on forces, inclines, friction

21 more projectiles

22 motion in a force field

Don't remove or overwrite **** or &&&&.  You and I both need those marks to be able to separate your answers from the question.  Your work would get less scrutiny if these marks aren't both present, and it will be harder for you when you want to review it.  If any of these marks are missing I will likely ask you to reinsert any missing marks and resubmit.

if rb has tension equal to weight of dominoes, what happens

if double; if fourple, etc.

rb force * dist, dist of slide

draw vectors representing init vel of ball off ramp, etc. etc. 

assign intro set 5; briefly introduce vectors in context of incline to show what they're good for

The Atwood machine

data

`q001.  Give your raw data for the Atwood machine.  This includes all directly observed quantities used in calculating your results.

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`q002.  For one of your trials show in detail how you use your raw data to obtain the acceleration of the system for that trial.

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`q003.  Give a table of acceleration vs. number of clips, one line at a time representing one trial at a time, with the number of clips then the acceleration for the trial separated by four spaces.  At the beginning or the end of the table, insert another line giving the units of each column.  Alternatively, you can copy a table made using a spreadsheet.

acceleration vs. number of clips:

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`q004.  Is it possible to fit a reasonable straight line to the data?  How much 'leeway' do you think you have in where to fit the line?

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`q005.  Give the coordinates of two points on the straight line you think comes as close as possible, on the average, to the points of your graph.  Use one point near each end of your line, rather that two points right next to one another.

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`q006.  Between the two points you specified in your preceding answer, what is the rise, what is the run and what therefore is the slope?  Be sure you specify the units of each of these quantities.

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`q007.  How plausible is it that the actual acceleration vs. number of clips is in fact well represented by a straight-line graph, with the deviations of the individual points from the straight line being due mostly to experimental uncertainties?

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`q008.  Specify the positive direction you chose for your system.  This can be specified by stating which mass goes which way in your chosen positive direction, or by stating whether the pulley rotates in the clockwise or counterclockwise direction when the system is moving in your chosen positive direction.

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`q009.  Suppose that each side of the pulley has a mass of 20 kg, and that each paperclip has a mass of about .2 kg (these masses are not realistic for the system we observed, being much greater than the masses we used in class).   Pretending that these are the actual masses in your system:

How much force is exerted by gravity on each side of the system when 1 paperclip is added to one side?

Assuming the absence of friction, what therefore is the net force on the system?

What is the mass of the system?

What therefore should be its acceleration?

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`q010.  Find the acceleration for the system in the preceding for 3 paperclips, and for 5 paperclips, added to the same side as before.

Sketch a graph of acceleration vs. number of paperclips and fit your best straight line to the graph.

How straight do you think your line is?

What is the slope of your line?  (be sure you include units)

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`q011.  For this series of examples, what is the mass of a single paperclip, as a percent of the mass of the entire system?

What percent of the acceleration of gravity is the slope of the graph you made for the preceding problem?

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`q012.  From the slope of the graph you made for your experiment, can you conjecture the mass of a paperclip as a percent of the mass of the system?

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`q013.  On your graph, what is the horizontal intercept of your straight line (i.e., if the line is extended, where does the line cross the x axis)? 

What are the units and the meaning of this point?

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`q014.  If the frictional force on the system is increased, would the acceleration of the system increase, decrease or stay the same?

What effect would this have on the points of your graph?

What effect would this have on the straight line that approximates your points?

What effect would this have on the x intercept of the straight line?

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Energy considerations

`q015.  Going back to the example problem where each mass is 20 kg and each clip has mass .2 kg, let's assume that three clips are added to the mass on the left, so that the system accelerates in the counterclockwise direction. 

We want to analyze the energy situation if the system moves .7 meters in our chosen positive direction.

What downward force is exerted on a 20 kg mass by gravity?

By how much does the gravitational potential energy of the 20 kg mass on the right therefore change as the system moves +.7 meters?

Answer the same for the 20 kg mass on the left.

Answer the same for the three clips.

What therefore is the PE change of the system?

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`q016.  Assuming that no nonconservative forces act on the system, what therefore is its change in kinetic energy?

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`q017.  The kinetic energy of the system is 1/2 m v^2, where m is the mass of the system.

Assuming it started from rest, its KE at the end of the interval will be equal to its change in KE.

What therefore is its KE at the end of the interval?

What is the mass of the system?

What therefore is its velocity at this point?

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`q018.  How would your answers to the last two questions change if there is a frictional force of 3 N acting on the system?

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vectors

`q019.  The figure on the board represented three vectors, one of magnitude 10 units at 305 deg, one of magnitude 8 units at 158 deg and one of 4 units at 80 deg.

According to our estimates:

• the x and y components of the first force are respectively 45% and -86% of that force
• the x and y components of the second force are respectively -90% and -46% of that force
• the x and y components of the third force are respectively 20% and 95% of that force

Based on these estimates calculate the x and y components of the three forces.

What is the sum of all the x components?

What is the sum of all the y components?

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`q020.  The actual percents are given by the sine and cosine functions as decimals.  For example if the percents are 45% and -86%, the cosine function would give us .45 and the sine function would give us -.86.

For an angle of 305 degrees, use your calculator to find sin(305 deg) and cos(305 deg).  (make sure you calculator is in 'degree' mode; using the sin/cos button find sin(305) and cos(305))

What do you get and how do the accurate values compare with our estimates?

Using the accurate values of the sine and cosine of 305 degrees, what are the x and components of the first vector?

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`q021.  Use the same procedure to find the x and y components of the second and third vectors.

Give you results below, and include a brief explanation of your results.

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`q022.  If you add up the x components of the three vectors, what do you get?

If you add up the y components of the three vectors, what do you get?

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Homework:

Your label for this assignment: 

ic_class_091014

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.

You should know everything in the first six problems of Set 5 in the Introductory Problem Set, which will give you a good, and not difficult, introduction to working with vectors.  A link that should get you there is at http://vhmthphy.vhcc.edu/ph1introsets/default.htm .