9 ANGULAR MOMENTUM 9.4 Angular momentum of a multi-component system
5Worked example 9.2: Angular momentum of a sphere
Question: A uniform sphere of mass M = 5 kg and radius a = 0.2 m spins about
an axis passing through its centre with period T = 0.7 s. What is the angular
momentum of the sphere?
Answer: The angular velocity of the sphere is
2 π
ω = =
T
2 π (^)
0.7
= 8.98 rad./s.
The moment of inertia of the sphere is
I =
2
M a^2 = 0.4 × 5 × (0.2)^2 = 0.08 kg m^2.
Hence, the angular momentum of the sphere is
L = I ω = 0.08 × 8.98 = 0.718 kg m^2 /s.
Worked example 9.3: Spinning skater
Question: A skater spins at an initial angular velocity of ω 1 = 11 rad./s with her
arms outstretched. The skater then lowers her arms, thereby decreasing her mo-
ment of inertia by a factor 8. What is the skater’s final angular velocity? Assume
that any friction between the skater’s skates and the ice is negligible.
Answer: Neglecting any friction between the skates and the ice, we expect the
skater to spin with constant angular momentum. The skater’s initial angular
momentum is
L 1 = I 1 ω 1 ,
where I 1 is the skater’s initial moment of inertia. The skater’s final angular mo-
mentum is
L 2 = I 2 ω 2 ,
where I 2 is the skater’s final moment of inertia, and ω 2 is her final angular veloc-
ity. Conservation of angular momentum yields L 1 = L 2 , or
ω =
I 1
ω.
(^2) I
2
2