9-1 Simple Rotations of a Rigid Body

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Transcript 9-1 Simple Rotations of a Rigid Body

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Physics for Scientists and
Engineers, 3rd edition
© 2005 Pearson Prentice Hall
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Chapter 9
Rotations of Rigid Bodies
Main Points of Chapter 9
• Rotation of a rigid body – angular velocity,
angular acceleration
• Rotational kinetic energy
• Rotational inertia
• Torque
• Angular momentum
• Conservation of angular momentum
• Rolling
9-1 Simple Rotations of a Rigid Body
• When a rigid body rotates
around a single axis, all
points on it move through
the same angle in the same
• Distance of the point from
the axis doesn’t change
9-1 Simple Rotations of a Rigid Body
Angular Velocity
• Simple rotations are onedimensional – the only
variable needed is the angle θ
• If the angular velocity ω is
constant, the angle increases
linearly with time
9-1 Simple Rotations of a Rigid Body
Right-hand rule for finding
direction of angular velocity:
9-1 Simple Rotations of a Rigid Body
For simple rotations, the equations of
motion are analogous to those for onedimensional linear motion:
9-1 Simple Rotations of a Rigid Body
Acceleration of a Point in a Rotating Rigid Body
If the angular velocity is increasing, there is a
tangential component to the acceleration as well as
the radial component.
9-2 Rotational Kinetic Energy
The kinetic energy of rotation can
be found by adding up the kinetic
energy of each piece of the object:
Kinetic energy of rotation:
This is made simpler by defining the
rotational inertia:
Rotational inertia:
9-3 Evaluation of Rotational Inertia
Rotational inertia of a continuous
object is found by integrating:
These integrals have been done
for many standard geometries.
9-3 Evaluation of Rotational Inertia
9-3 Evaluation of Rotational Inertia
Parallel-Axis Theorem
The parallel-axis theorem lets us calculate the
rotational inertia around any axis once we know
it around a parallel axis passing through the
center of mass.
Ipa is the rotational inertia around the parallel axis,
Icm the rotational inertia around the center of mass,
M the total mass, and d the distance from the
parallel axis to the center of mass.
9-3 Evaluation of Rotational Inertia
This shows an example of axes used in
the parallel-axis theorem:
9-4 Torque
What causes rotation?
• Force
• Lever arm – how far is force from
axis of rotation?
• Angle – does force have
components toward or away from
the axis of rotation?
9-4 Torque
The torque depends on the force, the
lever arm, and the angle:
The torque causes angular
acceleration; just how much
depends on the rotational inertia:
9-4 Torque
A right-hand rule
gives the direction of
the torque.
9-4 Torque
Gravity and Extended Objects
Gravitational torque acts at the center of mass, as
if all mass were concentrated there:
9-5 Angular Momentum and Its
Angular momentum is
analogous to linear momentum:
Torque is given by the change
in angular momentum:
Rotational kinetic energy
can also be written in terms
of angular momentum:
9-6 Rolling
Connection between speed and
angular velocity:
This assumes rolling without slipping.
9-6 Rolling
Motion is rotation around
contact point (with respect to
that point), giving the kinetic
We can use the parallel-axis
theorem to find Icontact:
9-6 Rolling
Equivalently, we can write the kinetic
energy of a rolling object as the sum of:
• the kinetic energy of rotation about its
center of mass, and
• the kinetic energy of the linear motion of
the object as if all the mass were at the
center of mass.
9-6 Rolling
Therefore, the rate at which a rolling
object accelerates depends on its
rotational inertia – this determines how
much of its kinetic energy will be
rotational and how much translational.
Summary of Chapter 9
• Rotations of a rigid body are one-dimensional
• Equations parallel those of linear motion
• If angular velocity is increasing, acceleration
has a transverse component
• Kinetic energy of rotation depends on
rotational inertia
• Parallel-axis theorem – once rotational inertia is
known about any axis, the rotational inertia
around any parallel axis can be found easily
Summary of Chapter 9, cont.
• Torque depends on force, lever arm, and angle
• Angular acceleration depends on torque and
on rotational inertia
• Torque is change in angular momentum
• Rolling without slipping: combination of
rotation and translation
• Rotational inertia determines how much
potential energy becomes translational kinetic
energy and how much rotational