The motion of rigid body

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review
The motion of rigid body
Combination of translation and rotation
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Rotation about a fixed axis
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Kinetic energy
Moment of inertia
Angular momentum
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Theorem of rotation
Eg. Find the kinetic energy of the earth ,given
its mass m, moment of inertia J, and orbit
velocity v .
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For a system of particles The change of total
energy equals the work done by internal
nonconservative force and external force
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?. 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
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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.
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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 .
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Solution 3.
dynamics
??
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The force exerted by the pivot
The motion of center of mass
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• 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
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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
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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 ?
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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.
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Theorem of angular momentum
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Under what conditions will the total angular
momentum of the system conserve ?
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When the resultant torque equals zero
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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?
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Change of kinetic energy for the system
Nonconservative internal force has done positive
work
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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
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collision
The momentum of the plaster transferred to the
angular momentum of the stick
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mechanical energy conserves for the system

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an athlete can change her spinning speed by
changing her posture, why?
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precession
definition
The revolution of the axis of spinning objects
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precession
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precession
L
dL
L
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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