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386 CHAPTER 12 Rotation of a Rigid Body

MOMENT OF INERTIA OF RIGID BODY 2

(see also Table 12.3) I R dm
where R is the radial distance of the mass
M
element dm from the axis of rotation; for dm dx
uniformly distributed mass, dm is given by [length]
M
dm dA, (12.33)
[area]

M
dm dV
[volume]

PARALLEL-AXIS THEOREM 2
I I Md (12.38)
CM
where M is the total mass and dis the distance from CM axis.

KINETIC ENERGY OF ROTATION K I 2 (12.28)
1
2

QUESTIONS FOR DISCUSSION

1. A spinning flywheel in the shape of a disk suddenly shatters 6. What configuration and what axis would you choose to give
into many small fragments. Draw the trajectories of a few of your body the smallest possible moment of inertia? The greatest?
these small fragments; assume that the fragments do not inter- 7. About what axis through the center of mass is the moment of
fere with each other. inertia of this book largest? Smallest? (Assume the book is
2. You may have noticed that in some old movies the wheels of closed.)
moving carriages or stagecoaches seem to rotate backwards. 8. A circular hoop made of thin wire has a radius R and mass M.
How does this come about? About what axis perpendicular to the plane of the hoop must
3. Relative to an inertial reference frame, what is your angular you rotate this hoop to obtain the minimum moment of inertia?
velocity right now about an axis passing through your center What is the value of this minimum?
of mass? 9. Automobile engines and other internal combustion engines
4. Consider the wheel of an accelerating automobile. Draw the have flywheels attached to their crankshafts. What is the
instantaneous acceleration vectors for a few points on the rim purpose of these flywheels? (Hint: Each explosive combustion
of the wheel. in one of the cylinders of such an engine gives a sudden push
5. The hands of a watch are small rectangles with a common axis to the crankshaft. How would the crankshaft respond to this
passing through one end.The minute hand is long and thin; push if it had no flywheel?)
the hour hand is short and thicker. Assume both hands have 10. Suppose you pump a mass M of seawater into a pond on a hill
the same mass. Which has the greater moment of inertia? at the equator. How does this change the moment of inertia of
Which has the greater kinetic energy and angular momentum? the Earth?

PROBLEMS

12.2 Rotation About A Fixed Axis † 2. Quito is on the Earth’s equator; New York is at latitude 41
north. What is the angular velocity of each city about the
1. The minute hand of a wall clock has a length of 20 cm. What Earth’s axis of rotation? What is the linear speed of each?
is the angular velocity of this hand? What is the speed of the
3. An automobile has wheels with a radius of 30 cm. What are
tip of this hand?
the angular velocity (in radians per second) and the frequency
(in revolutions per second) of the wheels when the automobile
†
For help, see Online Concept Tutorial 15 at www.wwnorton.com/physics is traveling at 88 km/h?