Mech#3 on 1974 AP "C" Exam -- 2 Blocks Connected by a Spring

[jgebhardt]

Homework Statement

A system of two blocks (each of mass M) are connected by a spring with spring constant k. The system is shoved to the left against a wall and the spring is compressed a distance D. The block(s) are then released. Determine the period of oscillation for the system when the left-hand block is no longer in contact with the wall.

Homework Equations

T=2π√m/k

The Attempt at a Solution

So my inclination is to say that T=2π√M/k since I assumed we could consider the oscillation of one individual block to be the same as the other. However, I have the solutions provided by the College Board and they indicate that the mass should be 1/2M resulting in T=2π√M/2k. Can someone explain why the mass is halved?

 

[TSny]

Hello. Welcome to PF!

What can you say about the motion of the center of mass of the system?

I assumed we could consider the oscillation of one individual block to be the same as the other.

Yes, that's true if you are considering the motion of each mass relative to the center of mass. Except, as one mass moves to the right relative to the CM, then which direction does the other mass move relative to the CM?

If one of the masses changes its distance from the CM by Δx, how much does the spring stretch?

 

[jgebhardt]

OK, so I get that there are multiple reference perspectives but I'm still unclear as to why that affects T since x is not part of the period equation. I understand that x is involved in finding k but since k is given...If I consider on block to be stationary and the other oscillating with respect to the stationary block I still would have a mass of M oscillating.
I hope I'm not being stupid here but I'm still perplexed. Further assistance?

 

[TSny]

You don't want to go the reference frame of one of the blocks since the blocks are accelerating and you would have a noninertial reference frame.

However, once the system loses contact with the wall, the CM reference frame is an inertial frame. So, you can analyze the motion of one of the masses in the CM frame. Does Hooke's law apply to each mass? Keep in mind that if the mass on the right moves Δx to the right, then the mass on the left must move -Δx (to the left). So, how much force acts on the mass on the right when it is displaced Δx from its equilibrium position?