2) Two marble spheres of masses 30 and 20 grams are suspended from the ceiling by massless strings. The lighter sphere is pulled aside through an angle of 75 degrees and let go. IT swings, collides elastically with the other sphere at the bottom of the swing.

a)to what maximum angle will the heavier sphere swing?

b) to what maximum angle will the lighter sphere swing?

This problem employs both conservation of momentum and conservation of energy.

First we need to find out how fast the lighter marble is moving when it strikes the heavier marble. So let's define the zero point for the gravitational potential energy to be the level where the two marbles collide. So we know that the lighter marble starts with a height of

\(\displaystyle h_0 = L - L~cos(75)\)

where L is the length of the string. (Notice that we don't know what this is.)

So using conservation of energy on the lighter marble:

\(\displaystyle K_0 + P_0 = K + P\)

\(\displaystyle mgh_0 = \frac{1}{2}mv^2\)

\(\displaystyle v = \sqrt{\frac{2L(1 - cos(75))}{g}} \approx 0.3889 \sqrt{L}\)

Now we can use conservation of momentum and conservation of energy (the collision is elastic) to find the speeds of both marbles after the collision. I won't go through the whole derivation, but we start with

\(\displaystyle m_1v_{01} + m_2v_{02} = m_1v_1 + m_2v_2\)

and

\(\displaystyle \frac{1}{2}m_1v_{01}^2 + \frac{1}{2}m_2v_{02}^2 = \frac{1}{2}m_1v_{1}^2 + \frac{1}{2}m_2v_{2}^2\)

and we can derive that

\(\displaystyle v_1 = v_{01} \frac{m_1 - m_2}{m_1 + m_2}\)

and

\(\displaystyle v_2 = 2v_{01} \frac{m_1}{m_1 + m_2}\)

when \(\displaystyle v_{02} = 0\)

So I get that

\(\displaystyle v_1 = - 0.07778 \sqrt{L}\)

and

\(\displaystyle v_2 = 0.31112 \sqrt{L}\)

using 1 as the lighter marble and 2 as the heavier marble. Don't worry about that negative sign on the lighter marble's velocity: that merely means that the marble has rebounded.

I'll leave it to you to finish up: it's just the reverse of the first step. And don't worry, that L will cancel itself out. If you have any troubles finishing, just let me know.

-Dan