these are in the 'preliminary notes' stage now; will be emailed and dated and unitalicized when completed

Ball off 60-cm ramp inclined at intervals of 1/24 inch.  Incline was changed by one turn of the 24-pitch screw for each trial.

At-home experiments:

Conservation of Energy on a Ramp:

Use an incline with a very small slope. 

Strike the steel ball so it coasts up, comes to rest, then accelerates back down the incline.  Let the ball continue to roll off the incline and fall to the floor, and mark the position at which it hits the floor.  (Be sure you also mark the straight-drop position).

Also note the position on the incline at which the ball comes to rest before accelerating back down. 

At the same time, using the TIMER program, record the time interval from the end of the 'strike' back to the end of the incline.

Repeat for at least two good trials.

Report your data:

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From each of the stopping points on the ramp, as you previously observed them, release the ball from rest and time it down the incline.  Record the positions at which it strikes the floor.

Report your data:

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From your data, infer how long it took the ball to go up the ramp for each trial.

Calculate the ball's acceleration up the ramp, and its acceleration down the ramp.

Report your results, and indicate how they were obtained:

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From your two accelerations, you can infer the coefficient of rolling friction.  What is your result?

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The large ball has diameter 2.5 cm, the small ball diameter 2.0 cm.  The ball is made of steel with an approximate density of 8 grams / cm^2. 

Find the KE the ball attained while rolling down the ramp with the PE it lost while rolling down.  Compare the two:

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Calculate the energy lost to rolling friction.  What do you get?

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Can you account for all PE that was lost as the ball rolled down the ramp?  Give details.

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Rotational dynamics with magnets on strap

You will accelerate the strap using a chain of rubber bands, as demonstrated in class.

Assume for now that the restoring force of your chain is

F = - k x,

where x is the length of the chain in excess of its length when supporting a single suspended domino, and k is

k = (1/2 N / cm) / (number of rubber bands in chain).

For example, suppose you have a chain of 7 rubber bands, which has length 44 cm when supporting the weight of a domino.  When the chain is 50 cm long, it exerts a force, which is calculated as follows:

For this chain k = ( 1/2 N / cm ) / 7, since there are 7 rubber bands.  We get k = .07 N / cm, approx..

The chain is 6 cm longer than it was when supporting a domino, so x = 6 cm.

The force exerted at this length is therefore F = - k x = -.07 N / cm * 6 cm = -.42 N.

Forces for other lengths can be calculated using F = - k x = -.07 N / cm * x.  Just remember that x is the length in excess of the 44 cm basic length.

Loop the last rubber band in your chain around the edge of the strap in such a way that when released the strap rotates in the clockwise direction.  Most likely this will cause the nut to spin in such a way that it descends the threaded rod. 

Extend the rubber band chain far enough that when the strap is released it will rotate through several revolutions before coming to rest.  However make it spin it fast enough to be dangerous, and make sure it's spinning slowly enough that you can count its revolutions.

Find a chain length that accomplishes the above goals. 

What is the length?

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You will use this chain length for every trial in your experiment.

Advice:

For each of the trials below, if the nut does rotate, return it to its original position between trials.

If you wish you can lubricate the system with WD-40 or something similar.  This is entirely optional.  If you don't have anything like WD-40 handy, don't worry about it.  Your instructor doesn't yet know whether this will yield better results, or worse results, than not lubricating the system.

Using this chain length for every trial, do five trials, releasing the system and counting the half-revolutions.  It should take several seconds for the system to come to rest, so you should also be able to look at the second hand or digital display of a clock or wristwatch when you release the strap, and when it comes to rest.  You should be able to remember the two clock positions and the number of half-revolutions long enough to write them down. 

Report your data for these five trials:

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Now repeat, but this time with a magnet halfway between the axis and each end of the strap.  Again do five trials.

Report your data for these five trials:

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Repeat once more, with the magnets near the ends of the strap.  Again do five trials.

Report your data for these five trials:

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Based on your data:

What is the median number of half-revolutions through which the strap rotated for each set of five trials?

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What is the median time required for the strap to come to rest for each set of five trials?

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Based on your median values, what is the average angular velocity of the strap for each set of five trials, calculated in half-revolutions / second?

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Assuming uniform angular acceleration, what is the initial angular velocity of the strap for each set of five trials, calculated in half-revolutions / second?

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Assuming uniform angular acceleration, what is the acceleration of the strap for each set of five trials, calculated in half-revolutions / second^2?

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The moment of inertia of the strap is about 1/12 m L^2, where L is its length (about 30 cm) and m its mass (about 65 grams).  The moment of inertia of a magnet whose center is at distance r from the axis is m r^2, where m is its mass (about 50 grams).

What therefore is the moment of inertia of the strap (plus magnets if present) in each of your trials?

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The angular kinetic energy of an object whose moment of inertia is I is given by the formula KE = 1/2 * I * omega^2, where omega is its angular velocity.  Since a half-revolution corresponds to pi radians of angular distance, the angular velocity in rad / sec is pi times the angular velocity in half-rev / sec. 

What is the angular KE of the strap corresponding to each set of five trials, based on the initial angular velocity you previously calculated?

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What average force was exerted by the rubber band system during each trial?  This result should be the same for all trials, if you followed the instruction to use the same chain length for each trial.

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Through what distance do you estimate the rubber band chain exerted its force?  Again a single result will probably apply to all trials.

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About how much work did the rubber band chain therefore do during each trial?  Once more a single result will probably apply to all trials.

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What is the average torque on the system during each trial?  (recall that torque = force * moment-arm; in this case the moment-arm is the distance from the axis to the point where the rubber band exerted its force)

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Through what fraction of a radian do you estimate the system traveled between release and the instant the rubber band chain went slack?

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What therefore is the product of average torque and angular displacement, for this interval?

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wd40

turn counterclockwise

... ball built of tiny magnets cleverly distributed on thin straps ...

... how to check for self-consistency

... edge effect

momentum

rotation ...