ROTATIONAL DYNAMICS

Busch apparatus using compressed air to minimize disk rotational friction (Douglass apparatus has a disk mounted on bearings)

Objective: To extend your understanding of conservation of momentum and energy from linear to angular. To understand the analogy of mass and moment of inertia.

Apparatus: Timer, Photogate, Air Disk Table or Rotating Disk with flag, Masses, Pulley

Introduction Rotating systems involve concepts that are similar to those in translational motion. The magnitude of rotational or angular velocity is given by

(1)

where v is the velocity that is perpendicular to the radius, r, from the center of rotation. As with linear velocity, angular velocity is a vector, but a more complicated “cross-product” vector formed from the two vectors r and v. We will measure only magnitudes

“Inertia” is the generic word for the “resistance” an object has to being accelerated. It is called the “moment of inertia” for rotational acceleration and depends upon the shape (geometry) of the rotating body and the location of the axis of rotation with respect to the center of mass. In the case of a single, compact mass particle, m, moving in a circle of radius, r, the moment of inertia is

(2)

For a solid disk the moment of inertia about a perpendicular axis through its center of mass, where mass=M and radius=R, is determined by adding the contributions of each particle in the disk according to their perpendicular distance to the axis. The net result is:

(3)

For translational motion the kinetic energy is defined as ½ mv_c.m.² where v is the linear velocity of the center of mass. The rotational kinetic energy is

where w is the angular velocity about the rotation axis. The total mechanical energy of a rotating and translating rigid object of mass, m, is the sum of translational kinetic energy, rotational kinetic energy, and translational potential energies. Relative to the center of mass, c.m. and an axis of rotation through the c.m.

where h is the height (gravitational PE), and the variation of g with height is ignored..

In an earlier experiment we examined the conservation of energy for a bucket (drive mass) falling some distance as the bucket pulled a low friction cart on a track. This demonstrated how the total mechanical energy of translational motion was conserved. The sum of the kinetic and potential energies of the glider and bucket was the same before and after the bucket fell (ignoring friction, pulley and wheel spin energy and motion of the string). The potential energy of the bucket was converted into kinetic energy of the bucket and cart. Now, we will examine the case for rotational motion.

The apparatus diagram shows a string of negligible mass wrapped around a disk initially at rest on an air table. The string is fed over a low friction and low moment of inertia pulley. On release, the pull of the hanging drive mass m₁ makes the disk rotate. The potential energy of the drive mass is converted into kinetic energy of translation (½ m₁v², of the drive mass) and rotational kinetic energy of the disk .

Releasing the disk from rest ( ), the speed of m₁ is the same as that of the disk rim: . The final angular velocity of the disk, calculated from the time nT_f for n revolutions immediately after m₁ hits the floor, thus gives the final velocity of m₁ just before it hit the floor and allows calculation of the final kinetic energies to check energy conservation.

Angular Momentum: In earlier experiments we observed how linear momentum is conserved when objects interact, if there is no net external force. For example, when two gliders collide on an air track they briefly exert equal and opposite forces on each other during the collision. The net (horizontal) force is zero and total momentum is conserved, though there is exchange of individual momentum. With gliders having magnetic bumpers that do not permanently absorb energy (convert organized mechanical energy into heat) the collision is called "elastic”, where both mechanical energy and momentum are conserved. In an inelastic collision, in which the gliders stuck together, momentum is conserved but not mechanical energy; some mechanical energy is lost to deformation. The low friction track experiment demonstrated conservation of translational (linear or straight-line) momentum.

In both translational (straight-line) and rotational motion, momentum is the product of an “inertia” and a “velocity”. In translational cases it is . Similarly, for its rotational counterpart, called angular momentum L, we have

or simply, L = Iw . (4)

L is angular momentum, I is “moment of inertia” and is the angular velocity in (rad/sec).

Angular momentum is conceptually similar to translational momentum, but the frame of reference is the axis of the rotational motion. Consider a swinging ball of mass m, tied to a string of length r. Its moment of inertia is mr²; its angular velocity is v/r. Thus, its angular momentum is where the length of string defines the radius, r, of the circular motion. Quickly shortening the string to reduce r results in the ball swinging around faster. v increases because r decreases. The angular momentum mvr remains constant. This is the same principle by which a figure skater makes herself spin faster. She gives herself a slow spin on one leg with her arms and the other leg stretched out. Bringing them inwards increases her spinning rate.

In the drawing below a low friction air table supports a single rotating disk of mass M and radius R. The timer is used to determine the time for n rotations nT, which is used to

calculate the angular velocity, . The initial angular momentum, L_i, is

the moment of inertia, I_i, times the initial angular velocity,. A small mass, m, is carefully dropped onto the rotating disk. Now, the disk with the mass rotates more slowly at final angular velocity,, because the new moment of inertia is now the total of the moments of inertia of the disk and of the mass, m.

A small mass m acquires angular momentum when dropped onto a spinning disk. Friction forces between mass and disk are internal, so angular momentum is exchanged between disk and mass, with conservation of the total. However, energy is dissipated in the work done by the friction.

The conservation of angular momentum is, mathematically

(5)

where initially, we have only the disk rotating:

(6)

and then we have the disk and dropped mass rotating (we treat the small mass as a point):

; (7)

Measurement of the initial and final angular velocities allows determination whether the angular momentum has been conserved: . Measurements must be made quickly to avoid the cumulative effect of friction that is always present.

You will analyze repeated measurements to test rotational energy conservation and angular momentum conservation in the two situations described above.

_Procedure:

_{Please Read Through and Understand This BEFORE Starting the Experiment. Experimental Technique is Very Important.}

1. First, run the experiment with just the gray disk. Record the mass, M, and the radius, R, of the disk; also, record the total mass of the mass hanger, m (try 50 - 150 g).

2. Turn on the R.U. timer, flip the switch to Timer and set the timer mode to BETWEEN PULSES (setting in photo below is “Pulse Low”), which gives measures the time interval of one complete disk rotation. The first pass of the flag through the photogate starts the timer; the second pass stops the timer. Note that the third pass again re-starts the timer, but starts adding to the time it recorded when it was stopped by the second pass. You will therefore have to press the reset button on the timer after the secon pass to record the correct time interval for the next cycle. This will be crucial during this experiment, since you will have the flag starting and stopping the timer several times (while the mass is accelerating) before the actual measurement is to be taken (after the mass hits the floor).

3. Position the photogate such that the paper flag taped onto the disk passes through the photogate as it comes around, as in picture below:

4. Hang the mass, m₁, over the pulley and carefully wrap the string around the aluminum shaft below the disk. The string must be long enough for it to still pull the disk until the drive mass strikes the floor. The string should fall free after the mass hits the floor. A finger on the top of the disk will hold everything in place.

5. Record the height of the drive mass before releasing the disk. Then remove your finger to release the disk and mass without applying any spin.

6. As soon as the drive mass strikes the floor and the string unwinds from the disk, press the reset button on the timer. Remember that you should press the reset button only when the timer is stopped. Read the time interval after it passes through the photogate . The timer will display the time it took for the disk to make one complete revolution.

6. Record your data for three trials at the same height. Use the average result for your calculations.

Angular Momentum Experiment:

In the picture above the low friction air table supports a single rotating disk of mass, M, and radius R. The timer is used to determine the time for one rotation, T, which is used to calculate the angular velocity, . The initial angular momentum, L_i, is the moment of inertia, I_i, times the initial angular velocity,. A small mass, m, is carefully dropped onto the rotating disk. Now, the disk with the mass rotates slower at final angular velocity,, because the new moment of inertia is now the total of the moments of inertia of the disk and of the mass, m.

_{Procedure: Please Read Through and Understand This BEFORE Starting the Experiment. Experimental Technique is Very Important.}

1. Make sure the flag trips the photgate as before, but also see to it that the mass you will drop onto the disk will not hit either the photogate or the flag – be ready to drop it at a different radius.

2. Record the mass, M, and the radius, R, of the disk; also, record the small mass, m₂ (try _~ 200 g).

3. Give the disk a slow spin such that it is rotating between one to two times a second. Trial and testing is needed to find the "right" speed for you.

4. Find the time for one rotation as you had done in the previous part of the lab; Then, QUICKLY do #5 before friction slows the disk. Later calculate T_i, the initial rotation time for one rotation.

5. Now, without disturbing anything, carefully drop the mass onto the disk from a small height of, say, 1-2 cm. Dropping the mass near the edge gives largest effect. (Why?).

6. Quickly, reset the timer and find the time for one rotation; record T_f (divide your time by the number of rotations if necessary).

7. Stop the rotating disk (turn off the air) without disturbing the mass, m. Measure r, the distance between the center of the mass, m, and the axis of rotation (rotation radius). Record your measurement and fill in your workshop report form. (We approximate m as a point mass, since it is small compared with the disk.)

8. After studying angular momentum, you will use these data to check whether mechanical energy was conserved. But for now, focus on the concept of angular momentum.

Rotational Dynamics - Hand-in Sheet

NAME: ____________________________________________Date: _____________

PARTNER(S): _________________________________________Section: ___________

PART 1: ROTATIONAL ENERGY

Drive mass, m: ___________

Mass of the "double-thick" disk, M: ____________ Radius of the disk, R: ____________

Starting height of drive mass, h: _______________

Angular Velocity:

Trial 1 Time ________ One Rotational Period, T=________

Trial 2 Time ________ One Rotational Period, T=________

Trial 3 Time ________ One Rotational Period, T=________

Average Rotational Period, T_av = _________

(use T_av for T) Final Angular velocity.  = _____________

The drive mass is "tied" to the rotating disk, so it falls with a final velocity that is related to the rotational velocity of the disk:

Final velocity of drive mass, v = _______________

is the radius around which the string is wrappeds it is the radius of the thick part of the shaft (not the lower thin part) [=1.5cm]. The string should come to the pulley tangent to the disc, fall free as it unwraps and not rewrap after the drivemass hits the floor. Wrap the loose end first.

Before you can start calculating the energies, you need the moment of inertia of the disk

I = _____________________

For the table below you must show your calculations A through E. Enter the results in the table. NOTE: Some entries are zero or can be ignored. Why? (Assume no friction in pulley or in the air flow.)

A. Potential Energy of the Drive Mass, m

INITIAL:

FINAL:

B. Potential Energy of the Disk, M

INITIAL:

FINAL:

C. Kinetic Energy of the Drive Mass, m

INITIAL:

FINAL:

D. Kinetic Energy of the Disk, M

INITIAL:

FINAL:

E. Total Energy

INITIAL:

FINAL:

Now enter the results of your calculations in this table:

How well was mechanical energy conserved, or nearly so?

PART 2: ANGULAR MOMENTUM

Mass of the disk, M: ______________ Radius of the disk, R: ____________

Mass of dropped small mass, m: _______________

Initial Angular Velocity: (If this is too great, the dropped mass will have insufficient friction and will spin off.)

# of Rotations: ________ Time _________ One Rotational Period, T_i = ___________

Initial Angular velocity, _i = __________

Final Angular Velocity:

# of Rotations: ________ Time _________ One Rotational Period, T_f = ___________

Final Angular velocity, _f = __________

Radial distance of small mass, m, from center of rotation, r: __________

A: Angular Momentum of Disk, M

INITIAL:

FINAL:

B. Angular Momentum of SMALL Mass, m

INITIAL:

FINAL:

C. Total Angular Momentum

INITIAL:

FINAL:

Show Results of Calculations in the Table:

Question #1: A theoretical value of the ratio, L_{f /} L_i, is 1.0 means perfect conservation of angular momentum. Why? Your result in all likelihood will be different. Discuss your results in 3 sentences. Why would L_{f /} L_i probably be less than 1, rather than greater?

Question #2: Do this qualitative experiment (you can use the timer if it helps): Spin the disk with the mass, m, already in place. Quickly pull off the mass, m, without hitting or disturbing the disk. Will the disk speed up or slow down? Try it and explain what you see in terms of what you learned in this lab. Justify your findings the best you can.

Now determine whether mechanical energy was conserved when the mass was dropped onto the rotating disk. (If dropped from just above the rotating disk, m will have little (negligible) initial and final potential energy, and zero initial kinetic energy. If dropped near the rim of the rotating disk, the final velocity of m will be * R_disk. You will need the moment of inertia of the disk and the dropped mass m.)

Some quantities are zero or can be ignored. Which? Show all calculations. Was mechanical energy conserved? . Was this an elastic or inelastic "collision"? Why or why not?

Summarize your theoretical expectations for the results of Part 2, for energy and for angular momentum conservation. (What objects does your “system” include? Consider friction in the slow-down process. Were the frictional forces internal or external to the “system”?)