III) A point mass m is attached to a string which passes through a hole in a table (see the Figure below). Mass m = 0.2 kg rotates in a circle of radius r = 0.8 m with an angular velocity of 40 rad/s. (a) What mass M must be attached to the end of the string under the table to maintain this motion? (b) Suppose that mass M is slowly increased by an amount that makes it descend a distance 0.1 m. What will the new angular velocity of the point mass be? (c) What is the amount of the increase of M? 30 pts.

a) Newton's second law gives us the following equations:

For the point mass m, .

For the hanging mass M, T-Mg=0.

Eliminating T from these two equations gives us

=26.1kg

b)

Conservation of angular momentum gives

c)

IV) What is the minimum kinetic energy needed to allow an unmanned rocket of mass 1200 kg to leave the moon's surface and arrive at a point very far away? Ignore the effect of the earth. 20 pts.