Problem 4.45

4.45. a)

First of all, notice that . These two are a force pair, so they must be equal in magnitude. Now, if we focus on the mass M, we see that the gravitational force exerted by the mass must be balanced by the force of tension in . Therefore,

Now, let's focus on the bottom pulley and we notice that

Now, by looking at the top pulley, we find that

Now, since the pulleys are both massless and frictionless, then it must be the case that

and

This leads us to the conclusion that

, so if we plug in what we know about , we find that

then we find

, and

b)

Since , we have .