Goal Apply conservation of angular momentum to a simplesystem.

Problem A student sits on a pivoted stool while holding a pair ofweights. (See Figure) The stool is free to rotate about a verticalaxis with negligible friction. The moment of inertia of student,weights, and stool is 2.25 kg Â· m2. The student is set in rotationwith arms outstretched, making one complete turn every 1.26 s, armsoutstretched.

Question If the student suddenly releases the weights, how does hisown angular momentum, rotational kinetic energy, and angular speedchange? (Select all that apply.)

His rotational angular speed decreases.
His rotational angular speed remains the same.
His angular momentum remains the same.
His rotational kinetic energy remains the same.
His angular momentum increases.
His angular momentum decreases.
His rotational angular speed increases.
His rotational kinetic energy increases.
His rotational kinetic energy decreases.

A star with an initial radius of 1.00 108 m and period of 30.0 dayscollapses suddenly to a radius of 1.00 104 m.

(a) Find the period of rotation after collapse.

(b) Find the work done by gravity during the collapse if the massof the star is 2.00 1030 kg.

The only force doing work on the system is gravitation, so the workdone by gravitation is the change in kinetic energy of the star.J

(c) What is the speed of an indestructible person standing on theequator of the collapsed star? (Neglect any relativistic or thermaleffects, and assume the star is spherical before and after itcollapses).

HINTS
* Do you expect the initial angular speed to be larger or smallerthan the final angular speed? How is angular speed related toperiod of rotation?

* Is the initial moment of inertia larger or smaller than the finalmoment of inertia?

* Is angular momentum conserved?

* As the star collapses, do you expect the period of rotation tobecome smaller or larger?

A playground ride consists of a disk of mass M = 56 kg and radius R= 2.0 m mounted on a low-friction axle. A child of mass m = 21 kgruns at speed v = 2.8 m/s on a line tangential to the disk andjumps onto the outer edge of the disk.

ANGULAR MOMENTUM
(a) Consider the system consisting of the child and the disk, butnot including the axle. Which of the following statements are true,from just before to just after the collision?
The momentum of the system doesn't change.
The momentum of the system changes.
The axle exerts a force on the system but nearly zero torque.
The torque exerted by the axle is zero because the force exerted bythe axle is very small.
The angular momentum of the system about the axle changes.
The angular momentum of the system about the axle hardlychanges.
The torque exerted by the axle is nearly zero even though the forceis large, because || is nearly zero.

(b) Relative to the axle, what was the magnitude of the angularmomentum of the child before the collision?
|C| = kgÂ·m2/s
(c) Relative to the axle, what was the angular momentum of thesystem of child plus disk just after the collision?
|C| = kgÂ·m2/s
(d) If the disk was initially at rest, now how fast is it rotating?That is, what is its angular speed? (The moment of inertia of auniform disk is Â½MR2.)
= radians/s
(e) How long does it take for the disk to go around once?
Time to go around once = s
ENERGY
(f) If you were to do a lot of algebra to calculate the kineticenergies before and after the collision, you would find that thetotal kinetic energy just after the collision is less than thetotal kinetic energy just before the collision. Where has most ofthis energy gone?
Increased chemical energy in the child.
Increased thermal energy of the disk and child.
Increased translational kinetic energy of the disk.

MOMENTUM
(g) What was the speed of the child just after the collision?
v = m/s
(h) What was the speed of the center of mass of the disk just afterthe collision?
vcm = m/s
(i) What was the magnitude of the linear momentum of the disk justafter the collision?
|p| = kgÂ·m/s
(j) Calculate the change in linear momentum of the systemconsisting of the child plus the disk (but not including the axle),from just before to just after impact, due to the impulse appliedby the axle. Take the x axis to be in the direction of the initialvelocity of the child.
px = px,f - px,i = kgÂ·m/s

ANGULAR MOMENTUM
(k) The child on the disk walks inward on the disk and ends upstanding at a new location a distance R/2 = 1 m from the axle. Nowwhat is the angular speed? (It helps to do this analysisalgebraically and plug in numbers at the end.)
= radians/s
ENERGY
(l) If you were to do a lot of algebra to calculate the kineticenergies in part (k), you would find that the total kinetic energyafter the move is greater than the total kinetic energy before themove. Where has this energy come from?
Decreased chemical energy in the child.
Decreased momentum of the disk.
Decreased thermal energy of the disk and child.

AST201H1 Lecture Notes - Lecture 3: Angular Momentum, Big Bang, Thermal Energy

AST201H1 Full Course Notes

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