Calculate New RPM: Spinning Rate of 4g Bug on 12g Disk with 30cm Radius

[joeypeter]

A bug that has a mass mb = 4 g walks from the center to the edge of a disk that is freely turning at 21 rpm. The disk has a mass of md = 12 g. If the radius of the disk is R = 30 cm, what is the new rate of spinning in rpm?

Can I have some help please?

 

[kkrizka]

Try using conservation of anugular momentum. Also remember that moments of inertia add, so the moment of the inertia for your system is the moment of disk + moment of bug.

 

[Moetasim]

Sure! To calculate the new rate of spinning, we can use the conservation of angular momentum equation:
Initial angular momentum = Final angular momentum
Since the bug walks from the center to the edge of the disk, the initial angular momentum is mb x ωi x Ri, where mb is the mass of the bug, ωi is the initial angular velocity (in radians per second) and Ri is the initial distance of the bug from the center of the disk. Similarly, the final angular momentum is (mb + md) x ωf x Rf, where ωf is the final angular velocity (in radians per second) and Rf is the final distance of the bug from the center of the disk (which is equal to the radius of the disk, R).
Setting these two equal to each other, we get:
mb x ωi x Ri = (mb + md) x ωf x R
Rearranging for ωf, we get:
ωf = (mb x ωi x Ri) / ((mb + md) x R)
Plugging in the given values, we get:
ωf = (0.004 kg x (21 rpm x 2π/60 s) x 0.3 m) / ((0.004 kg + 0.012 kg) x 0.3 m)
Simplifying, we get:
ωf = 14.2 rad/s
Converting to rpm, we get:
New rate of spinning = 14.2 rad/s x (60 s/2π) = 136.4 rpm
Therefore, the new rate of spinning is 136.4 rpm.