Rotation of a Rigid Body

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Rotation of a Rigid Body Readings
Chapter 13
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How can we characterize the acceleration during
rotation? - translational acceleration and -
angular acceleration
Newtons second law
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Angular acceleration
Center of rotation
Both points have the same angular velocity

Linear acceleration
Both points have the same angular acceleration
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Rotation of Rigid Body
Every point undergoes circular motion with the
same angular velocity and the same angular
acceleration
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The relation between angular velocity and angular
acceleration is the same as the relation between
linear velocity and linear acceleration
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The Center of Mass
For Rigid Body sometimes it is convenient to
describe the rotation about the special point
the center of mass of the body.
Definition The coordinate of the center of mass
Rigid body consisting of two particles
If then
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The Center of Mass
Definition The coordinate of the center of mass
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The Center of Mass Example
The center of mass of a disk is the center O of
the disk
O
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Torque Rotational Equivalent of Force
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Torque
The rotation of the body is determined by the
torque
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Torque
Torque is maximum if Torque is 0 if
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Torque
Torque is positive if the force is trying to
rotate the body counterclockwise Torque is
negative if the force is trying to rotate the
body clockwise
axis
The net torque is the sum of the torques due to
all applied forces
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Torque Example
Find the net torque
axis
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Torque Relation between the torque and angular
acceleration
- moment of inertia
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Moment of Inertia
Thin rod, about center
Thin rod, about end
Cylinder (or disk), about center
Cylindrical loop, about center
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Moment of Inertia Parallel-axis Theorem
If you know the moment of Inertia about the
center of mass (point O) then the Moment of
Inertia about point (axis) P will be
P
O
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Parallel-axis Theorem Example
Thin rod, about center of mass
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Equilibrium
Massless rod
Two forces (which can results in rotation) acting
on the rod
Equilibrium
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Rotational Energy
Conservation of energy (no friction)
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Kinetic energy of rolling motion
Cylinder
Cylindrical loop
Solid sphere