Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lectures 35-37

Hw: Chapter 15 problems and exercises

An ant of mass m is standing at the center of a massless rod of length l. The rod is pivoted at one end so that it can rotate in a horizontal plane. The ant and the rod are given an initial angular velocity 0. If the ant crawls out towards the end of the rod so that his distance from the pivot is given by , find the angular velocity of the rod as a function of time, angular momentum, force exerted on the bug by the rod, torque about the origin.

2

2bt

l

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.

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

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

For symmetrical objects rotating about their axis of symmetry:

2);( ii

irmIrhrIL

)(rhrIext

Second Law:

)(2

1 22VVmKE r

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?

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?

A man stands on a massless platform that is free to rotate in the horizontal plane. He holds a weight in each hand that has mass m. He has his arms extended so that they have length S. The system is set into rotation so that the angular velocity of the platform is ω0.

Assume the man’s mass can be neglected compared to the weights. What force would have to be applied to one of the weights at the distance S so that in t0 seconds the platform, which is initially at rest, is rotating with angular velocity ω0?

Assuming the man’s mass can be neglected compared to the weights, what would be the angular velocity of the system if he starts with angular velocity ω0 and then brings his arms in so that the distance from his center is reduced from S to S/4 ?

A vertical axle is free to rotate. A massless rod is attached to the axle, as shown, and there are two masses, m1 and m2 attached to the rod. The axle is given an angular velocity ω0. Some internal spring-like force ejects mass m2 so that it leaves the rod perpendicular to the rod, horizontally, with velocity of magnitude v1. Neglect gravity.a. What will be the angular velocity of the rod after the mass is ejected?

b. What would the angular momentum of the rod be if instead of being massless it had a moment of inertia I, about the axle?

A cylinder rolls down an inclined plane of angle θ without slipping. a) Find the acceleration of the cylinder.b) Find the maximum θ if the coefficient of static friction is μ.

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.

Gyroscopic precession

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

A rotating flywheel

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 ;

€

V = Vr2 +Vθ

2

)(][

;

;

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