Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lectures 24, 25

Hw: Chapter 15 problems and exercises

Suppose there were an axle at the origin with a rigid, but massless rod attached to it with bearings so that the rod could freely rotate. At the end of the rod, of length b, there is a block of mass M as shown below:

b

x0

v0, mrod

axle

A bullet is fired at the block. If the bullet strikes the block and sticks, what will be the angular velocity of the block about the axle? Neglect gravity.

A block of mass M is cemented to a circular platform at a distance b from its center. The platform can rotate, without friction, about a vertical axle through its center with a moment of inertia, Ip. If a bullet of mass m, moving horizontally with velocity of magnitude vB as shown, strikes and imbeds itself in the block, find the angular velocity of the platform after the collision.

b

top view

vB

axle

Ex. 4A platform can rotate, without friction, about a vertical axle through its center with a moment of inertia, Ip. A small bug of mass m is placed on the platform at a distance b from the center, and the system is set spinning with angular velocity 0 (Clockwise as viewed from above).a) What is the total angular momentum of the system with the bug at rest on the platform?b) What is the total angular momentum if he runs in the opposite direction to the platform’s rotation?c) Is it possible for a little bug to stop the big platform from rotating?

Two disks are spinning on a frictionless axle. The one on the left has mass M, radius RL, and moment of inertia about the axle IL. It is spinning with L in the direction shown. It is moving to the right with velocity of magnitude v0. The one on the right has mass M, radius RR, and moment of inertia about the axle IR. It is spinning with R in the direction shown. It is moving to the left with velocity of magnitude v0. When the two disks collide they stick together. What is the velocity and the angular velocity of the combined system after the collision?

A man stands on a platform which is free to rotate on frictionless bearings. He has his arms extended with a huge mass m in each hand. If he is set into rotation with angular velocity 0 and then drops his hands to his sides, what happens to his angular velocity? (Assume that the man’s mass is negligible and that his arms have length R when extended and are R/4 from the center of his body when at his sides.)

What is the moment of inertia of a disk of thickness h, radius R and total mass M about an axis through its center?

For symmetrical objects rotating about their axis of symmetry:

2);( ii

irmIrhrIL

)(rhrIext

Second Law:

)(2

1 22VVmKE r

m1

m2

R I

The rope is assumed not to slip as the pulley turns. Given m1, m2, R, and I find the acceleration of mass m1.

Rotational Kinetic Energy

2

2

1 IKE

22

2

1

2

1 cmcm ImvKE

A rigid body in motion about a moving axis

Motion of a rigid body: combination of translation motion of the center of mass and rotation about an axis through the center of mass.

22

2

1

2

1 cmcm IMvK

A primitive yo-yo is made by wrapping a string several times around a solid cylinder with mass M and radius R. You hold the end of the string stationary while releasing the cylinder with no initial motion. The string unwinds but does not slip or stretch as the cylinder drops and rotates. Use energy considerations to find the speed vcm of the center of mass of the solid cylinder after it has dropped a distance h.

Consider the speed of a yo-yo toy

The yo-yo (again)

Gyroscopic precession

• The precession of a gyroscope shows up in many “common” situations.

A rotating flywheel

Ex. 7

Find the minimum value of the coefficient of friction in order for a cylinder to roll without slipping down an inclined plane of angle θ.

Conservation of Momentum

extFdt

pd

)()(

)()(

,0,0

afterpbeforep

afterpbeforep

ConstpConstp

Constpdt

pdFIf

yy

xx

yx

ext

If the collision is perfectly elastic, the kinetic energy is conserved!

Circular Motion

rdt

drar

dt

rda

maFmaF

r

rr

2;

;

22

2

y

x

r

ri

i

rirr

r

dt

drV

dt

drVr ;

)(][

;

;

2 rhrmriridt

drmrprL

dt

Ld

prL

Fr

r

exttot

Torque and Angular Momentum

Conservation of Angular Momentum

Ra

For symmetrical objects rotating about their axis of symmetry:

2);( ii

irmIrhrIL

)(rhrIext

Second Law:

)(2

1 22VVmKE r

Have a great day!

Hw: Chapter 15 problems and exercises