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Chapter 8
Rotational Equilibrium and Rotational Dynamics
1.
Torque
2.
Torque and Equilibrium
3.
Center of Mass and Center of Gravity
4.
Torque and angular acceleration
5.
Rotational Kinetic energy
6.
Angular momentum
7.
Conservation of angular momentum
Torque
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What is torque?
How do I calculate it?
What are its SI units?
How do is compare to force?
How do I find the direction of torque?
How do I add two or more torques?
Torque
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But wait, what does the torque equation
really mean?
Lever Arm
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What is a lever arm?
How does it help?
Right Hand Rule
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Point the fingers
in the direction of
the position
vector
Curl the fingers
toward the force
vector
The thumb points
in the direction of
the torque
Right Hand Rule
1.
A fishing pole is 2.00
m long and inclined to
the horizontal at an
angle of 20.0° (Fig.
P8.6). What is the
torque exerted by the
fish about an axis
perpendicular to the
page and passing
through the hand of
the person holding the
pole?
Torque and Equilibrium
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Example - Equilibrium
1.
A uniform horizontal 300-N
beam, 5.00 m long, is attached
to a wall by a pin connection
that allows the beam to rotate.
Its far end is supported by a
cable that makes an angle of
53.0° with the horizontal. If a
600-N person stands 1.50 m
from the wall, find the tension
in the cable and the force
exerted by the wall on the
beam.
Axis of Rotation
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If the object is in equilibrium, it does
not matter where you put the axis of
rotation for calculating the net torque
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The location of the axis of rotation is
completely arbitrary
Often the nature of the problem will suggest
a convenient location for the axis
When solving a problem, you must specify
an axis of rotation
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Once you have chosen an axis, you must
maintain that choice consistently throughout the
problem
Center of Gravity
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What is center of
gravity?
How do I calculate
it?
Is there an easier
way?
What about
arbitrary objects?
Example - Center of Gravity
1.
Find the center of gravity for
the 3 mass system shown in
the figure.
Moment of Inertia
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What is moment of Inertia?
How do I calculate it?
What are its SI units?
Moment of Inertia of a
Uniform Ring
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Other Moments of Inertia
Torque and Angular
Acceleration
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Newton’s Second Law for
a Rotating Object
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How do I write Newton’s second
law for rotating rigid bodies?
Example, Newton’s Second
Law for Rotation
1.
A solid, frictionless cylindrical
reel of mass M=3 kg and
radius R=0.4 m is used to
draw water from a well. A
bucket of mass m=2 kg is
attached to a cord that is
wrapped around the cylinder.
If the bucket starts from rest
at the top of the well and falls
for 3.0 s before hitting the
water, how far does it fall?
Rotational Kinetic Energy
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How do I calculate it?
What are the SI units?
Total Energy of a System
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Conservation of Mechanical Energy
Example - Rotational Kinetic
Energy
1.
A sphere and a cylinder rolls
down an inclined plane of height
h. Which object reaches the
bottom first?
Work-Energy in a Rotating
System
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Example - Work-Energy in
a Rotating System
Attached to each end of a thin steel rod of length 1m
and mass 6.2 kg is a small ball of mass 1.10 kg. The
rod is constrained to rotate in a horizontal plane
about a vertical axis through its midpoint. At a certain
instant, it is rotating at 39.0 rev/s, because of friction,
it slows to a stop in 32 s. Assume a constant frictional
torque.
1.
a)
b)
c)
d)
Compute the angular acceleration
Compute the retarding torque due to friction
Compute the total energy transferred from mechanical
energy to thermal energy by friction
Compute the number of revolutions rotated during 32 s.
Angular Momentum
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What is angular momentum?
How do I calculate it?
What are the SI units?
How do I relate it to torque?
What about conservation?
Example - Angular
Momentum
1.
A student sits on a rotating stool
holding two 3.0-kg objects.
When his arms are extended
horizontally, the objects are 1.0
m from the axis of rotation, and
he rotates with an angular speed
of 0.75 rad/s. The moment of
inertia of the student plus stool
is 3.0 kg • m2 and is assumed to
be constant. The student then
pulls the objects horizontally to
0.30 m from the rotation axis.
(a) Find the new angular speed
of the student. (b) Find the
kinetic energy of the student
before and after the objects are
pulled in.
Example - Angular
Momentum: Neutron Star
1.
During a supernovae explosion a stars core
collapses from a radius of R=1.0x104km
and an initial period of rotation of 30 days
to R=3km. Find the new period of rotation
of the star’s core.