m 2 . are Group Answer the following Questions. (2x8 =16) 6. An ice skater, such as the one in Figure alongside, is spinning at 0.800 rev/ s with her arms extended. She has a moment of inertia of 2.34 kg m² with her arms extended and of 0.363 kg · m² with her arms close to her body. (These moments of inertia based on reasonable assumptions about a 60.0-kg skater.) a. What is her angular velocity in revolutions per second after she pulls in her arms? (2) b. What is her rotational kinetic energy before and after she does this? (2) c. Why is final kinetic energy more as you calculate in part (6) even if final moment of inertia is decreased? Explain. (2) d. State principle of conservation of angular momentum and explain why does she stretch her arms out when she wants to come to rest? (2)
Answers
Answer:
So far, we have looked at the angular momentum of systems consisting of point particles and rigid bodies. We have also analyzed the torques involved, using the expression that relates the external net torque to the change in angular momentum, (Figure). Examples of systems that obey this equation include a freely spinning bicycle tire that slows over time due to torque arising from friction, or the slowing of Earth’s rotation over millions of years due to frictional forces exerted on tidal deformations.
However, suppose there is no net external torque on the system,
∑
→
τ
=
0.
In this case, (Figure) becomes the law of conservation of angular momentum.
Law of Conservation of Angular Momentum
The angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net external torque around that point:
d
→
L
d
t
=
0
or
→
L
=
→
l
1
+
→
l
2
+
⋯
+
→
l
N
=
constant
.
Note that the total angular momentum
→
L
is conserved. Any of the individual angular momenta can change as long as their sum remains constant. This law is analogous to linear momentum being conserved when the external force on a system is zero.
As an example of conservation of angular momentum, (Figure) shows an ice skater executing a spin. The net torque on her is very close to zero because there is relatively little friction between her skates and the ice. Also, the friction is exerted very close to the pivot point. Both
|
→
F
|
and
|
→
r
|
are small, so
|
→
τ
|
is negligible. Consequently, she can spin for quite some time. She can also increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that
L
′
=
L
or
I
′
ω
′
=
I
ω
,
where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because
I
′
is smaller, the angular velocity
ω
′
must increase to keep the angular momentum constant