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review

The motion of rigid body

Combination of translation and rotation

Rotation about a fixed axis

Kinetic energy

Moment of inertia

Angular momentum

Theorem of rotation

Eg. Find the kinetic energy of the earth ,given

its mass m, moment of inertia J, and orbit

velocity v .

For a system of particles The change of total

energy equals the work done by internal

nonconservative force and external force

?. Work and energy in rotation

For rigid body

Gravitational potential energy of rigid body

The gravitational potential energy of rigid body

equals that of the center of mass

When dWe dWi,n0, Ek Ep constant

eg. A uniform stick can rotate around a

horizontal axis freely.

1. calculate its angular velocity when it falls

from the horizontal position.

2. the force exerted by the pivot.

Solution 1.

Work and energy

For the stick

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Solution 2

Conservation of energy

The mechanical energy of the system for the stick

and the earth conserves .

Solution 3.

dynamics

??

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The force exerted by the pivot

The motion of center of mass

- 5.5 conservation of angular momentum

angular momentum for a particle

Angular momentum of rigid body

Theorem of angular momentum

For a system if Mdt 0, angular momentum conserves

Eg .two flywheels will stick together ,their

former angular velocity are

1.Find their final angular velocity

2.The loss of mechanical energy

There is no external torque

The total angular momentum of the system conserves

Loss of kinetic energy

The loss of mechanical energy is due to the work

done by the nonconservative internal force

friction

If the two flywheels have the same moment of

inertia and have angular momentum along the same

direction , will there still be loss of

mechanical energy ?

Eg. Two flywheels will stick together,their

former angular velocity are

1. Find their final angular velocity

does the total angular momentum of the system

conserve?

With torque of the friction , one will accelerate

and the other decelerate.

Theorem of angular momentum

Under what conditions will the total angular

momentum of the system conserve ?

When the resultant torque equals zero

eg. The moment of inertia for the rotational

system is J0

The mass of the object hold by the person is m,

when the person draws in her arms from r1 to r2

1. find the angular velocity

2. The loss of mechanical energy

What is the total torque ?

The angular momentum of the system conserves.

What has changed during the process?

Change of kinetic energy for the system

Nonconservative internal force has done positive

work

eg . A stick with mass M and length l can

rotate freely about a horizontal axis , it

initially stays in the vertical position until a

piece of plaster with mass m and velocity v bumps

onto it , then they swing together .

find 1. The final angular velocity 2.

The biggest angle the stick can reach

The process can be further divided into two

subprocess

collision

m

Rotation about a fixed axis

M

collision

The momentum of the plaster transferred to the

angular momentum of the stick

mechanical energy conserves for the system

an athlete can change her spinning speed by

changing her posture, why?

precession

definition

The revolution of the axis of spinning objects

precession

precession

L

dL

L

F

v

F

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Eg. A flywheel with mass of m,its moment of

inertia is J, it is revolving with angular

velocity of ? in the horizontal plane , the

length of the stick is d,find the angular

velocity of the fly wheel about its axis .

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Homework

5.6 5.7 5.8