Chapter 8
The Dynamics of Rotation
Moment of Inertia
Torque with the moment of inertia
Moments-of-Inertia |
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Slender Rod |
Slender Rod |
Disk |
Cylinder |
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Rectangular |
Solid Sphere |
Hoop |
Hoop |
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Spherical Shell |
Cylinder |
Cone |
Hoop |
Example 8-18
Use the general definition of the moment of inertia to find the moment of inertia for the dumbbell shaped object shown below. Note the axis of rotation goes through the center of the object and points out of the page. In addition, assume that the masses may be treated as point masses.
Example 8-19
The motor in an electric saw brings the
circular blade from rest up to the rated angular velocity of 80.0 rev/s in
240.0 rev. One type of blade has a moment of inertia of 
.
What net torque (assumed constant) must the motor apply to the blade?
Example 8-20
The figure shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.
Rotational Kinetic Energy
The rotational kinetic energy 
of
a rigid object rotating with an angular speed 
about
a fixed axis and having a moment of inertia I is
The rotational work 
done
by a constant torque 
in
turning an object through an angle 
is
Total Mechanical Energy
Angular Momentum
The angular momentum L of a body rotating
about a fixed axis is the product of the body's moment of inertia I and
its angular velocity with
respect fo that axis:
Example 8-21
A thin-walled hollow cylinder (mass = 
,
radius = 
)
and a solid cylinder (mass = 
,
radius = 
)
start from rest at the top of an incline. Both cylinders start at the same
vertical height 
.
All heights are measured relative to an arbitrarily chosen zero level that
passes through the center of mass of a cylinder when it is at the bottom of
the incline. Ignoring energy losses due to retarding forces, determine which
cylinder has the greatest translational speed upon reaching the bottom.
Example 8-22
For a classroom demonstration, a sits on
a piano stool holding a sizable mass in each hand. Initially, the student holds
his arms outstretched and spins about the axis of the stool with an angular
speed of 3.74 rad/s. The moment of inertia in this case is 
.
While still spinning, the student pulls his arms in to his chest, reducing
the moment of inertia to 
.
What is the student's angular speed now?
Example 8-23
A star of radius 
rotates
with an angular speed 
.
If this star collapses to a radius of 20.0 km, find its final angular speed.
(Treat the star as if it were a uniform sphere, and assume that no mass is
lost as the star collapses.)
Example 8-24
A 34.0 kg child runs with a speed of 2.80
m/s tangential to the rim of a stationary merry-go-round. The merry-go-round
has a moment of inertia of 
and
a radius of 2.31 m. When the child jumps onto the merry-go-round, the entire
system begins to rotate. What is the angular speed of the system?
The End