ROTATIONAL DYNAMICS

Equipment: Timer, Photogate, Air Disk Table or Rotating Disk, Masses, Pulley

Reference: Physics, Cutnell and Johnson, 5^th ed., Chapters 8 & 9 (Wiley)

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

where (1)

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 below 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, drive mass) and rotation .

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. More detailed procedures for experiment and analysis will be available in class.

Preliminary Question Name/Course-section/Date _________________________

1. The earth has mass m_e = 5.98x10²⁴ kg. It orbits around the sun with frequency of one cycle per year at mean distance R_E = 1.496x10¹¹ meters.

Calculate the earth’s orbital angular momentum about the sun (mks), using this mean radius. L_{E orbital} = ________

The gravitational force of the sun maintains the earth in orbit. It is always directed toward the sun. The orbit is not quite circular, it is a (closed) ellipse. Does the earth’s orbital angular momentum vary with its distance from the sun? Explain your answer, in terms of torque on the earth due to the gravitational force of the sun.

The moment of inertia about an axis through the center of mass of a uniform, rigid sphere of radius R is 2/5 m _R². The radius of the earth (neglect the equatorial bulge due to spin frequency of one per day) is R_E = 6.37x10⁶ meters.

What would be the earth’s spin angular momentum¹, if it were of uniform density?

L_{E,rigid spin} = ______ (mks units)

Does the earth’s density increase or decrease toward the center? (Circle one.) Is it solid?

How are the answers to these questions determined experimentally?

Consistent with your answer about density variation with depth, and assuming the entire earth spins with the same angular velocity, will the actual L_{E spin} be greater than or less than the value you calculated above for a uniform density, rigid earth? (Circle one.) Explain.

2. It is possible to determine whether an egg has been boiled from its behavior when spun on a table. Describe the difference between raw and hard boiled eggs which you expect to observe. Explain in terms of the interior properties.