class 091202

Class 091130 draft

TIMER program

Do the experiment Using the Timer Program , down to the point where you see the instruction that reads as follows (you do not need to follow this instruction):

'You got at least 20 time intervals. Based on your data what was the average of the first 20 time intervals? Note that you could get this average by averaging the first 20 intervals. My first few intervals were .15625, .15625, .1875, .171875, etc; I could just add up the first 20 intervals and divide by 20 to get the average. However there is an easier and quicker way to get the result, so use the easier way if you can.'

Once you get the TIMER program to work, you should be able to run the gravitational simulation program from yesterday's class notes.

Experiment

Using a paperclip to fix the end of a thread in place, wrap the thread at least half a turn around the rim of the foam wheel. Suspend a mass from the other end of the thread, so that when the system is released the mass will be free to descend, with the resulting torque accelerating the wheel.

The figure below indicates a wheel with a blue string wrapped around its rim, from the position of the radial vector r_0 to the position of the radial vector r_hat, then extending a ways downward to a suspended mass. The system is assumed to rotate about an axis through the center of the circle.

The nondescript green shape represents a paperclip attached to the wheel in such a way that the thread will be anchored to the rim of the wheel. The thread will remain anchored as the wheel rotates through the r_hat position, but will slip off long after the wheel passes that position. The figure below shows a typical position at which the thread slips off: This event (the thread slipping off the clip, which separates the thread and suspended mass from the wheel) will be referred to as 'separation'.

The radial vector at this position is labeled r_sep. (standing for 'radial vector at separation').

As the wheel rotates beyond the r_hat position the thread will lose contact with the rim of the wheel but will stay in contact for a short time with the paperclip. The moment arm used to calculate the torque will therefore decrease a bit, but as long as the thread slips off somewhere around the position depicted in the second figure the moment arm will not have decreased too much. So as a first approximation, we can assume that the average torque is

tau_ave = -r_hat * suspended weight.

You will set the system up and observe angular position of the r_0 vector. Rest the axis on a relatively low-friction surface (you can just rest it on your fingers if you wish). When you release the system it will accelerate until reaching the r_sep position, and then coast to rest.

Observe the time required for the system to coast to rest
Observe the angle at which separation occurs.
Observe the angle through which the wheel rotates while coming to rest.
Measure the diameter of the wheel.
Note the mass of the suspended object.

`q001. Find the work done on the system by gravity, from the r_sep position to the position at which the wheel comes to rest. If you can solve all or part of this problem without following the prompts below, you may give your solution here. Otherwise answer after the prompts:

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What is the torque exerted by the gravitational force on the suspended mass?

Through what angle did the wheel turn while this torque was present?

Use your results to find the work done on the system by the suspended mass.

`q002. After the thread slips off the paper clip the system coasts to rest. Consider the interval which extends from the r_sep position until the wheel comes to rest.

Using your data, find the initial angular velocity omega_init of the wheel during this interval (we use 'omega_init' rather than 'omega_0' to avoid confusion with the r_0 notation, since r_0 refers to the position of the radial vector at release, which is not the beginning of the current interval).

Assuming that the angular acceleration of the system is uniform, find the angular acceleration of the system during this interval,

`q003. We consider again the interval between the r_0 and r_sep positions, the interval on which weight of the descending mass exerts a torque on the wheel. You previously calculated the work done by this torque.

If we assume that friction does negligible work on the system on this interval, then by the work-energy theorem, what should be the KE of the system at the r_sep position?

You now have sufficient information to determine the moment of inertia of the system. If you can solve all or part of this problem without following the prompts below, you may give your solution here. Otherwise answer after the prompts:

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Set the symbolic expression 1/2 I omega_init^2 equal the symbol KE, then solve the symbolic equation for I. What is your result?

You have values for KE and omega_init at the instant of separation. What therefore must be the moment of inertia of the wheel?

`q004. Determine the PE change of the suspended mass, between the release of the system from rest and the instant of separation. If you can solve all or part of this problem without following the prompts below, you may give your solution here. Otherwise answer after the prompts:

What is the angular difference between the r_0 and r_hat positions?

What length of thread was originally in contact with the rim of the wheel?

How far did the mass therefore descend between those two positions?

How much further did the mass descend before the instant of separation?

By how much did the PE of the descending mass therefore change between the instant of release and the instant of separation?

Escape velocity

Consider a projectile 'shot' at high velocity from the surface of the Earth in the radial direction. That is, the projectile is 'shot' directly away from the center of the Earth. We ignore the effect of the atmosphere, which in reality makes this impractical, and we assume that the projectile encounters no forces other than the gravitational force exerted by the Earth.

As the projectile moves away from the Earth, its gravitational PE increases and its KE therefore decreases. The only force acting on it is the force exerted by the Earth, so it encounters no nonconservative forces. It follows from energy conservation that `dPE + `dKE = 0.

We consider the interval between being 'shot' from the surface and coming to rest for an instant before returning to Earth. If the projectile keeps going without returning, then the interval will never end; but in either case the interval begins with the 'shot'.

We will let M be the mass of the Earth, m the mass of the projectile, R the radius of the Earth, and r the distance of the projectile from the center of the Earth.

`q005. Evaluate the right-hand side of the equation v_0 > = sqrt( 2 G M / R). What is your result?

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If v_0 is greater than your result (even a very little bit greater), then KE + PE > 0 at launch. If KE + PE > 0 at launch, then the same is true until the projectile experiences a nongravitational force. The only force it experiences is that of the Earth, so unless it returns to and crashes into the Earth it will never experience a nongravitational force. Until this happens, if ever, it remains the case that KE + PE > 0.

Can the projectile's PE (recall that in this analysis PE is measured relative to infinite separation from the Earth) ever be positive?

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Since for this case, with v_0 greater than your result, we have KE + PE > 0, is it possible for the projectile's velocity ever to be zero?

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Is it possible for this projectile to ever return?

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`q006. Suppose you are on a merry-go-round, riding a painted pony at a distance of 4 meters from the axis of rotation. What is your contribution to the moment of inertia of the merry-go-round?

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If the merry-go-round requires 12 seconds to complete a revolution, how fast are you moving and what is your angular velocity?

What is your kinetic energy, based on 1/2 m v^2?

What is your kinetic energy, based on 1/2 I omega^2?

Suppose the merry-go-round started from rest and required 20 seconds to get up to its present rate of rotation. Assuming uniform angular acceleration:

What was the angular acceleration?

How much of the torque produced by the motor was required to accelerate your mass?

How much force exerted in your direction of motion did it take to get your mass up to its present speed?

What is the moment arm from the axis of rotation to your position? If you multiply the force exerted in your direction of motion by this moment arm, what is your result?

`q007. You should be able to sketch each of the following, as sketched and discussed in class. Give a brief synopsis of the important characteristics of each sketch.

v vs. t graph trapezoid
vector projection (project one vector on another)
vector components on x-y coordinate system (sketched in standard vertical-horizontal position, and in 'rotated' positions)
forces on a ball on an incline without friction
forces on a free pendulum at a position away from equilibrium
addition of two vectors by sketching, adding and resolving components
mass on an incline incline with friction and a suspended mass
atwood machine
v vs. t and x vs. t graphs, interrelationships
circular motion with r, v, a_cent
forces in elliptical orbit
forces on domino on accelerating strap
SHM model
wheel with descending mass

Homework:

Your label for this assignment:

ic_class_091118

Copy and paste this label into the form.

Answer the questions posed above.

You have already seen most of the ideas in the qa's and Introductory Problem Set mentioned below. If you work through these documents as assigned, you will get plenty of practice and should develop good expertise with these concepts.

Do qa's #34 on moments of inertia and 35 on SHM.

Introductory Problem Set 9 consists of 17 problems on simple harmonic motion. You will be expected to work through these problems by the first of next week. http://vhmthphy.vhcc.edu/ph1introsets/default.htm .