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

Question

A kg figure skater is spinning on the loss of her skates at 1.5 rev/s. Her arms ate 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, 61 cm tall) plus two 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-tall cylinder. What is her new rotation frequency, in revolutions per second?

Explanation / Answer

The concept dealing here is the Conservation of Angular Momentum (which is same as the conservation of momentum, but which is in the rotational world). here Angular Momentum (L) can be obtained using:

L(final) = L(initial)
I * w (final) = I * w (initial)

Here I = moment of inertia of the object
w = angular velocity of the object.

We need to find her moment of inertia for each case:

Case 1:
Her body act as a cylinder spinning around its central axis: radius = 0.1 m, mass = 40kg.

Now moment of inertia of the cylinder is:

I = m * r2 / 2= I = 0.2 [kg * m^2]

Her arms will act as a rods: mass = 2.5kg, length = 0.61m.Now the moment of inertia of the rod

I = m * L^2 / 12 ==> I = 0.0775 [kg * m^2]

her arms is not rotating in its perpendicular axis, but it is rotating at the center of her body at some distance away. Which can be expressed as half the length of her arm + the radius of her body:

By parallel axis theorem, we can find the moment of inertia of each arm:

I = I(center) + m * D^2
I = 0.0775 + 2.5 * 0.40^2 ==> I = 0.4775 [kg * m^2]

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

I = I(body) + 2 * I(arms)
I = 0.2 + 2 * (0.4775) == > I = 1.155 [kg * m^2]

So, using the equation for angular momentum, we know her initial momentum is:

L(initial) = I * w
L = 1.155 [kg * m^2] * 1.5 [rev/s]
using conservation of momentum, we know the final momentum= original momentum. So, first we need to find her new moment of inertia.

Case 2:
now she act as one big cylinder: radius = 0.1 m, mass = 45kg:

I = m * r^2 / 2 ==> I = 0.225 [kg * m^2].

And finally:

L(final) = L(initial)
I * w (final) = 1.155 (from before)

w = 1.155 / I
w = 1.155 / 0.225 ==> w = 5.13 rev/s
Her New Rotation frequency , w = 5.13 rev/s

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