Classical Mechanics, Coupled Harmonic motion

[ForMyThunder]

Homework Statement

Set up the equations of motion for the system shown in Fig. 4.16. The relaxed lengths of
the two springs are l1, l2 . Separate the problem into two problems, one involving the motion
of the center of mass, and the other involving the "internal motion" described by the two
coordinates x1, x2. Find the normal modes of vibration.

Homework Equations

Figure 4.16 is attached

The Attempt at a Solution

The problem is already turned in and we have a test tomorrow but here is the general outline of my solution. So the center of mass of blocks 1 and 3 is

r_{13}= xm_3+(x-l_1-x_1)m_1

and using Newton's law, and the fact that the force on this subsystem is -k_2x_2, we have

(m_1+m_2)\ddot r_{13}= m_3\ddot x+(\ddot x-\ddot x_1)m_1=-k_2x_2

and similarly with the system 2-3

the total force on just block 3 is easy so we have these three equations, assume x, x_1, and x_2 all have the form Ce^pt for DIFFERENT constants C. Substitute and solve for p.

Main question: are the equations I outlined above correct?

 

[BruceW]

The variables are x,x_1,x_2 so I would define each of them to be positive going towards the right. You haven't done that, so it makes it a bit more confusing, but still technically correct.

You should try the form Ce^pt only after you have taken away the motion of the centre of mass. (I'm guessing you knew that, but they might give you marks for saying it explicitly, so you should probably say it).