(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A circular disc of radius 25 cm and mass 0.5 kg is revolving in its plane with an angular velocity of 4 radians per second. Find A) its kinetic energy of rotation, and B) its new angular velocity if a mass of 10 kg is suddenly fixed on the rim of the disc.

2. Relevant equations

E_{k}= 1/2 I ω^{2}

Moment of inertia for disc = I = 1/2 mr^{2}

Parallel axis theorem: I_{o}= 1/2 mr^{2}+ mr

Conservation of angular momentum: L_{i}= L_{f}

3. The attempt at a solution

A)

I = 1/2 mr^{2}

I = 1/2 (0.5 kg) (0.25 m)^{2}= 0.015625 kg m^{2}

Kinetic energy of rotation = = 1/2 I ω^{2}

Kinetic energy of rotation = 1/2 (0.015625)(4)^{2}= 0.125 J

B)

The new mass shifts the axis of rotation from the center point of the disc to a point O on the rim of the disc. The distance between the two points is equal to the radius (0.25 m). Parallel axis theorem applies:

I_{o}= moment of inertia on the new point on the rim of the disc

I_{o}= 1/2 mr^{2}+ mr

I_{o}= 1/2(0.5 kg)(0.25 m)^{2}+ (0.5 kg)(0.25 m)

I_{o}= 0.140625 kg m^{2}

Since there are no external torque acting of the disc, conservation of angular momentum applies. So, initial angular momentum is equal to final angular momentum.

I * ω_{i}= I_{o}* ω_{f}

(0.0156 kg m^{2})(4 rad/s) = (0.140625 kg m^{2}) ω_{f}

ω_{f}= 0.44 rad /s

The new angular velocity is 0.44 rad /s

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I did my best to solve the problem as well as I could. Is my reasoning correct?

One thing I am not sure at all about is the effect of the mass of 10kg that was used to basically fix a new point of rotation on the disc. The way I understand it is that all that new mass does is just shift the axis of rotation to a new point, so we apply the parallel axis theorem to get the new moment of inertia. So the calculations would be the same if the mass was 20kg or 50kg, etc.

I appreciate any help!

Thanks

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# Homework Help: Kinetic energy of rotation and parallel axis theorem problem

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