Chapter 11

Lecture 19:

Angular Momentum: II

HW7 (problems): 9.61, 9.75, 10.2, 10.13, 10.30, 10.47, 10.51, 10.63

Due on Friday, Mar. 27.

Moments of Inertia of Various Rigid Objects

11.8: Newton’s 2nd Law in Angular Form

The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.

Sample problem: Torque, Penguin Fall Calculations: The magnitude of l can be found by using

The perpendicular distance between O and an extension of vector p is the given distanceD. The speed of an object that has fallen from rest for a time t is v =gt. Therefore,

To find the direction of we use the right-hand rule for the vector product, and find that the direction is into the plane of the figure. The vector changes with time in magnitude only; its direction remains unchanged.(b) About the origin O, what is the torque on the penguin due to the gravitational force ?

Calculations:Using the right-hand rule for the vector product we find that the direction of t is the negativedirection of the z axis, the same as l.

Angular Momentum of a Rotating Rigid Object

Each particle of the object rotates in the xy plane about the z axis with an angular speed of

The angular momentum of an individual particle is Li = mi ri

2 and are directed

along the z axis

L

Angular Momentum of a Rotating Rigid Object, cont

To find the angular momentum of the entire object, add the angular momenta of all the individual particles

This also gives the rotational form of Newton’s Second Law

2z i i i

i i

L L m r I

extzdL d

I Idt dt

Angular Momentum of a Rotating Rigid Object, final

The rotational form of Newton’s Second Law is also valid for a rigid object rotating about a moving axis provided the moving axis:(1) passes through the center of mass(2) is a symmetry axis

If a symmetrical object rotates about a fixed axis passing through its center of mass, the vector form holds: where is the total angular momentum measured with

respect to the axis of rotation

IL

L

Angular Momentum of a Bowling Ball

The momentum of inertia of the ball is 2/5MR 2

The angular momentum of the ball is Lz = I

The direction of the angular momentum is in the positive z direction

Conservation of Angular Momentum

The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero Net torque = 0 -> means that the system is isolated

For a system of particles,

tot i f = constant or = L L L

tot n = = constantL L

Conservation of Angular Momentum, cont

If the mass of an isolated system undergoes redistribution, the moment of inertia changes The conservation of angular momentum requires

a compensating change in the angular velocity Ii i = If f = constant

This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system

The net torque must be zero in any case

Conservation Law Summary

For an isolated system -

(1) Conservation of Energy: Ei = Ef

(2) Conservation of Linear Momentum:

(3) Conservation of Angular Momentum: (demo)

i fp p

i fL L

Af_1103.swf

11.11.3. Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close to his body and he rotates faster. Why does his speed increase?

a) By bringing the dumbbells inward, Joe exerts a torque on the stool.

b) By bringing the dumbbells inward, Joe decreases the moment of inertia.

c) By bringing the dumbbells inward, Joe increases the angular momentum.

d) By bringing the dumbbells inward, Joe increases the moment of inertia.

e) By bringing the dumbbells inward, Joe decreases the angular momentum.

11.11.3. Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close to his body and he rotates faster. Why does his speed increase?

a) By bringing the dumbbells inward, Joe exerts a torque on the stool.

b) By bringing the dumbbells inward, Joe decreases the moment of inertia.

c) By bringing the dumbbells inward, Joe increases the angular momentum.

d) By bringing the dumbbells inward, Joe increases the moment of inertia.

e) By bringing the dumbbells inward, Joe decreases the angular momentum.

11.11.4. Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both dumbbells. How does the rotational speed of stool change, if at all?

a) The rotational speed increases.

b) The rotational speed decreases, but Joe continues to rotate.

c) The rotational speed remains the same.

d) The rotational speed quickly decreases to zero rad/s.

11.11.4. Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both dumbbells. How does the rotational speed of stool change, if at all?

a) The rotational speed increases.

b) The rotational speed decreases, but Joe continues to rotate.

c) The rotational speed remains the same.

d) The rotational speed quickly decreases to zero rad/s.

11.11: Conservation of Angular Momentum

If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system.

(a) The student has a relatively large rotational inertia about the rotation axis and a relatively small angular speed.

(b) By decreasing his rotational inertia, the student automatically increases his angular speed. The angular momentum of the rotating system remains unchanged.

Sample problem

Figure 11-20 a shows a student, sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia Iwh about its central axis is 1.2 kg m2. (The rim contains lead in order to make the value of Iwh substantial.) The wheel is rotating at an angular speed wh of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum Lwh of the wheel points vertically upward. The student now inverts the wheel (Fig. 11-20b) so Lwh that, as seen from overhead, it is rotating clockwise.

Its angular momentum is now - Lwh.The inversion results in the student, the stool, and the wheel’s center rotating together as a composite rigid body about the stool’s rotation axis, with rotational inertia Ib= 6.8 kg m2. With what angular speed wb and in what direction does the composite body rotate after the inversion of the wheel?

Sample problem

Motion of a Top The only external forces

acting on the top are the normal force and the gravitational force

The direction of the angular momentum is along the axis of symmetry

The right-hand rule indicates that the torque is in the xy plane

M r F r g

Motion of a Top, cont The net torque and the angular momentum are

related:

A non-zero torque produces a change in the angular momentum

The result of the change in angular momentum is a precession about the z axis

The direction of the angular momentum is changing The precessional motion is the motion of the symmetry axis

about the vertical The precession is usually slow relative to the spinning

motion of the top

d

dt

L

Gyroscope

A gyroscope can be used to illustrate precessional motion

The gravitational force produces a torque about the pivot, and this torque is perpendicular to the axle

The normal force produces no torque

11.12: Precession of a Gyroscope

(b) A rapidly spinning gyroscope, with angular momentum, L, precesses around the z axis. Its precessional motion is in the xy plane.

(c) The change in angular momentum, dL/dt, leads to a rotation of L about O.

(a) A non-spinning gyroscope falls by rotating in an xz plane because of torque .

Gyroscope in a Spacecraft

The angular momentum of the spacecraft about its center of mass is zero

A gyroscope is set into rotation, giving it a nonzero angular momentum

The spacecraft rotates in the direction opposite to that of the gyroscope

So the total momentum of the system remains zero

New Analysis Model 1

Nonisolated System (Angular Momentum) If a system interacts with its environment in the

sense that there is an external torque on the system, the net external torque acting on the system is equal to the time rate of change of its angular momentum:

totd

dt L

New Analysis Model 2

Isolated System (Angular Momentum) If a system experiences no external torque from

the environment, the total angular momentum of the system is conserved:

Applying this law of conservation of angular momentum to a system whose moment of inertia changes gives

Iii = Iff = constant

i fL L