1. A merry-go-round spins without friction. The merry-go-round is a circular disk of mass M and radius R, and that it is initially spinning with an angular velocity ω0. Snow begins to fall at a constant rate of r kg/sec. Find an expression for the angular velocity of the merry-go-round as a function of time.
2. A circular disk with a non-uniform mass density is spinning at an angular frequency of 8 s-1 about its axis without friction. The disk has a thickness of 1 cm. Measurements of its mass density at several radii are given in the table below.
|
Radius (m)
|
Mass Density (kg/m3)
|
|
0
|
2500
|
|
0.01
|
2450
|
|
0.025
|
2250
|
|
0.035
|
2000
|
|
0.04
|
1750
|
|
0.045
|
1500
|
|
0.05
|
1250
|
|
0.065
|
750
|
|
0.085
|
400
|
|
0.10
|
200
|
a) Calculate the moment of inertia for the disk
b) After the disk has been spinning for a long time, it expands to twice its original radius but its thickness and total mass don't change. Assuming that all parts of the disk expand (for example, the total mass originally between 0 and 0.01 m is now spread out between 0 and 0.02 m) find a method to estimate the new moment of inertia. Prove that your total mass is essentially unchanged. Find the new angular frequency.