And must be measured about the same origin the

and 𝐿 must be measured about the same origin.

The average torque is given by Ԧ 𝜏 =
Δ 𝐿
Δ𝑡

This is the rotational analog of N2L in the form
σ
Ԧ
𝐹 =
𝑑 Ԧ𝑝
𝑑𝑡
.

For the rotation of a symmetrical object about the
symmetry axis, the angular momentum and
angular velocity are related by:
𝐿 = I 𝜔

49

Can an object moving in a straight line have nonzero angular
momentum?
Quick Quiz RB-8

50

Can an object moving in a straight line have nonzero angular
momentum?
Quick Quiz RB-8

51

Conservation of Angular Momentum
52

If the net external torque acting on a
system is zero, its angular momentum is
conserved:

A change in I for an isolated
symmetrical object rotating about its
symmetry axis implies a change in
angular speed:

Examples: Stellar explosions (neutron
stars), figure skaters, divers, …
). L L
f i
0   

(if
f
f
i
i
f i
ω I ω I
L L



53

For an object moving in a straight line with a constant velocity, is
the angular momentum conserved?
a)
Yes.
b)
No.
Quick Quiz RB-9

54

For an object moving in a straight line with a constant velocity, is
the angular momentum conserved?
a)
Yes.
b)
No.
Quick Quiz RB-9

Rotations Block 3 – Q1
55
An ice figure skater starts out spinning at 0.85 revolutions per
second with her arms outstretched. She wears lightly weighted
bracelets to enhance the spin-up effect when she pulls her arms in.
a)
Calculate her final rotational speed if her moment of inertia is
3.6 kg.m
2
with her arms outstretched and 1.1 kg.m
2
with her
arms pulled close to her body.
b)
Determine the increase in her rotational kinetic energy.
c)
Where does this added energy come from?

56
A disk with moment of inertia I
1
rotates about a frictionless, vertical
axle with angular speed 𝜔
𝑖
. A second disk, this one having moment of
inertia I
2
about the same axle and initially not rotating, drops onto the
first disk. Because of friction between the surfaces, the two eventually
reach the same angular speed 𝜔
𝑓
.
a) Calculate 𝜔
𝑓
.
b) Calculate the ratio of the final to the initial rotational energy.
Rotations Block 3 – Q2

57
A merry-go-round is a common piece of playground equipment. A 3.0-
m-diameter merry-go-round with a mass of 250 kg is spinning at 20
rpm. John runs tangent to the merry-go-round at 50 m/s, in the same
direction that it is turning, and jumps onto the outer edge. John’s mass is
30 kg.
a) What is the merry-go- round’s angular velocity, in rpm, after John
jumps on? You can treat the merry-go-round as a cylinder and John
as a point particle.
b) What happens to the angular velocity of the merry-go-round and
John if John now starts walking towards the center?
Rotations Block 3 – Q3