# Estimate the mass of a spinning ice skater

## Homework Statement

If you have data such as time of rotation when arms are extended and the lengths of their folded and extended arms measured from the centre of the skater.

I know I1ω1=I2ω2 and can work out ω from the radius and time of rotation. Where can I go from here to estimate the mass?

Thanks

## Homework Equations

$I_1\omega_1=I_2\omega_2$
$\omega = \frac{2\pi}{T}$

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gneill
Mentor
Create a crude model of the skater using simple geometric shapes for which you know (or can look up) the mass moment of inertia about various axes. For example, make the torso a cylinder of uniform density, the arms two more cylinders when outstretched. Estimate what you think might be reasonable geometry for them (length, diameter). When the arms are folded in, the arm cylinders just add their mass to the torso cylinder and otherwise disappear. You may need to use the parallel axis theorem to get the right center of rotation for your "arms".

The unknowns will be the mass of the torso cyclinder and the mass of an arm.

Create a crude model of the skater using simple geometric shapes for which you know (or can look up) the mass moment of inertia about various axes. For example, make the torso a cylinder of uniform density, the arms two more cylinders when outstretched. Estimate what you think might be reasonable geometry for them (length, diameter). When the arms are folded in, the arm cylinders just add their mass to the torso cylinder and otherwise disappear. You may need to use the parallel axis theorem to get the right center of rotation for your "arms".

The unknowns will be the mass of the torso cyclinder and the mass of an arm.

Would the expressions below be along the right lines?
$I_{in} = \frac{1}{2}*M_{torso}*r_{torso}^2 + m_{arms}*(r_{torso}+r_{folded arms})^2$
$I_{out} = \frac{1}{2}*M_{torso}*r_{torso}^2 + m_{arms}*(r_{torso}+r_{outstretched arms})^2$

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gneill
Mentor
That's the idea. You'll want to show in detail how you arrived at the expressions for the arms, the outstretched arms in particular. Check the mass moment of inertia for a cylinder rotating about an axis that is not parallel to its central axis (for example, suppose it were rotating about one end).

That's the idea. You'll want to show in detail how you arrived at the expressions for the arms, the outstretched arms in particular. Check the mass moment of inertia for a cylinder rotating about an axis that is not parallel to its central axis (for example, suppose it were rotating about one end).
Thank you for the help so far.

How would I go about using the above expressions to obtain a result for the torso mass and arm masses together?

gneill
Mentor
Presumably you'll apply your data and conservation of angular momentum to arrive at an equation for which you can solve for the mass. Another hint: It might be worthwhile assuming that the mass of the arms is some fraction of the mass of the torso, so you'll have only one mass variable to deal with.