A kg figure skater is spinning on the toes of her skates at 1.5 rev/s. Her arms

Question

A kg figure skater is spinning on the toes of her skates at 1.5 rev/s. Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 kg, 20 cm average diameter, 160 cm tall) plus rod like arms (2.5 kg each, 61 cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20 cm diameter, 200 cm fall cylinder. What is her new rotation frequency, in revolutions per second?

Explanation / Answer

Here we have to find the new rotation frequency when the skater raises her arm straight above her head.

The principle based on which, we will solve this is the " conservation of angular momentum".

This means in the absence of any external effects, the angular momentum of the skater before raising her arm is equal to the angular momentum after raising her arm.

i. the angular momentum of the skater before raising her arm

angular momentum= moment of inertia x angular speed

in this case, the skater is modelled as cylinder torso(40 kg, 20cm average diametre and 160 cm tall) plus two rod like arms ( 2.5 kg, 61cm long) attached to the outside torso.

so, Momemt of inertia of the skator = MI of the cylinder + MI of rods

= 1/2 x mass x radius2 + 1/3 x mass x length2 x 2

= 1/2 x 40 x 0.1 2 + 1/3 x 2.5 x 0.712 x 2 ( we have taken 0.71 instead of 0.61 because 0.71 is the length from the axis of rotation)

=0.2 + 0.84

=1.04 kg.m2

Angular momentum = 1.04 x 1.5 x 2 x pi ( 1.5 is not the angular velocity but it is the rotation frequency)

ii. i. the angular momentum of the skater after raising her arm

Momemt of inertia of the skator = MI of the cylinder

= 1/2 x mass x radius2

= 1/2 x 45 x 0.1^2

=0.225 kg.m2

Now if w = new rotation frequency

then angular momentum = 0.225 x w x 2 x pi

as angular momentum is conserved,

0.225 w= 1.5 x 1.04

so, w= 6.93 rev/s or 6.9 rev/s

so new angular frequency = 6.9 rev/s

all the best in the course work

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