v = rω
atan = rα
arad = rω2
a  a a
Rotation about a Moving Axis
• Every motion of of a rigid body can be represented
as a combination of motion of the center of mass
(translation) and rotation about an axis through the
center of mass
• The total kinetic energy can always be represented as
the sum of a part associated with motion of the center of
mass (treated as a point) plus a part asociated with
rotation about an axis through the center of mass
Total Kinetic Energy
Ktotal = (1/2)Mvcm2 + (1/2)Icmω2
A plumbing problem
Torque on a pulley
Unwinding a winch
Rotation about a moving axis
Conservation of angular momentum
When the sum of the torques of all external
forces acting on a system is zero, then
The professor as figure skater?
•It seems that danger to the instructor is proportional to interest in
any given demonstration.
Internal forces
cannot change
the total
momentum of a
Balancing on a teeter-totter –
Figure 10.25
• The
child must
sit closer
to balance
the torque
from the
•Refer to
10.11 on
page 314.
Balanced forces during exercise
A gyroscope in the laboratory
A 1.31-kg bowling trophy is held at arm’s length, a
distance of 0.505 m from the shoulder joint. What
torque does the trophy exert about the shoulder if
the arm is (a) horizontal, or (b) at an angle of 20°
below the horizontal?
A school yard teeter-totter with a total length of
5.2 m and a mass of 36 kg is pivoted at its center.
A 18 kg child sits on one end of the teeter-totter.
(a)Where should a parent push vertically downward
with a force of 210 N in order to hold the teetertotter level?
(b)Where should the parent push with a force of
310 N?
(c)How would your answer to parts (a) and (b) change
if the mass of teeter-totter were doubled?
Ch 9 Problem 39
The flywheel of a gasoline engine is required to give up
500 J of kinetic energy while its angular velocity
decreases from 650 rev/min to 520 rev/min. What
moment of inertia is required?
Ch 9 Problem 49
A size-5 soccer ball of diameter 22.6 cm and mass
426 g rolls up a hill without slipping, reaching a
maximum height of 5.00 m above the base of the hill.
We can model this ball as a thin-walled hollow sphere.
(a) At what rate was it rotating at the base of the hill?
(b) How much rotational kinetic energy did it then have?
Ch 9. Problem 51
A solid uniform cylinder and a solid uniform sphere,
each with the same mass and diameter, approach a
hill rolling with a forward speed of 6.50 m/s. Both of
them roll up the hill without slipping.
(a) Find the maximum height that each of the centers will
(b) Why do they reach different heights? Didn’t both of
them have the same speed at the bottom of the hill?
(c) Would your answer to part (a) be different if these
two objects (1) did not have the same masses or (2)
did not have the same diameter? What makes you
say this?
Ch 10 Problem 27
A certain drawbridge can be modeled as a uniform
15,000 N bar, 12.0 m long, pivoted about its lower
end. When this bridge is raised to an angle of
60.0 degrees above the horizontal, the cable
holding it suddenly breaks, allowing the bridge to
fall. At the instant after the cable breaks,
(a) what is this torque on the bridge about the pivot
(b) at what rate is its angular momentum changing?