A 45 kg figure skater is spinning on the toes of her skates at 1.2 rev/s . Her a

Question

A 45 kg figure skater is spinning on the toes of her skates at 1.2 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 two rod-like arms (2.5 kg each, 69 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 -tall cylinder.

What is her new rotation frequency, in revolutions per second?

Explanation / Answer

By Using Conservation of Angular Momentum (L)
L(final) = L(initial)
I * w (final) = I * w (initial)

Where I is the moment of inertia of the object
and w is the angular velocity of the object.

When her body is a cylinder spinning around its central axis: radius = 0.1 m, mass = 40kg.
I = m * r2 / 2

I = 0.2 [kg * m2]

Her arms are like rods: mass = 2.5kg, length = 0.69m.you can find the moment of inertia of a rod rotated about its perpendicular axis:

I = m * L2 / 12

I = 0.0991875 [kg * m2]

D = 0.69m / 2 + 0.1

D = 0.445 m

moment of inertia of each arm:
I = I(center) + m * D2

I = 0.0991875 + 2.5 * 0.4452

I = 0.59425 [kg * m2]

This is the moment of inertia for one arm. We already have the moment of inertia for her body, so we can add the body plus two arms to get her total moment of inertia:

I = I(body) + 2 * I(arms)

I = 0.2 + 2 * (0.59425) = 1.3885 [kg * m2]

L(initial) = I * w

L = 1.3885 [kg * m2] * 1.2 [rev/s]

She is now one big cylinder: radius = 0.1 m, mass = 45kg:

I = m * r2 / 2 = 0.225 [kg * m2].

L(final) = L(initial)

0.225 * w (final) = 1.3885 * 1.2

w = 7.40 rev/s

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