Momentum problem involving spring and possible harmonic motion PLEASE HELP

[fmilner]

Homework Statement

Two blocks mass m1 and m2 (m1 greater than m2) are joined by a spring (which can extend and compress) and both rest on a horizontal frictionless table. the blocks are pulled apart, held at rest and then released. describe and explain as fully as you can the sibsequent motion of the blocks (a well labelled velocity time graph would be suitable)

Homework Equations

P=mv F=kΔl

The Attempt at a Solution

 

[voko]

What is your difficulty with this problem?

 

[fmilner]

this a question i was set, and I'm not sure how exactly how to describe the displacement time graph, what the zero value should be or whether to think of the blocks separately or joined? would the motion be aysymmetric as one weight more than the other, as the force inward would relate to the position in the extension so acceleration a=f/m would be different for each and would the resulting momentum of the heavier block mean it compresses the spring more on one side
or am i over complicating this/

 

[voko]

Have you heard of the center of mass?

 

[fmilner]

yes...

 

[voko]

So think of this system in terms of its center of mass. Does it move? Think about the distances and velocities of the two masses relative to the center of mass. Are they related in any way?

 

[HallsofIvy]

I personally wouldn't use the "center of mass" (a moral failing, no doubt). I would, instead, set up two equations, one for each block. Let x1 be the disance from some fixed point, in line with the line between the two blocks, to block of mass m1 and x2 the distance from the fixed point to the block of mass m2. We can, without loss of generality, that m2> m1.

Then the distance between the two blocks is m2- m1 and so m1(d^2x1/dt^2)= -k(x2-x1) while m2(d^2x2/dt^2)= k(x1- x2). That is a pair of first order differential equations which can be solved in either of two ways.

1) Differentiate the first equation to get a second derivative of x2, then use the second equation to eliminate x2, giving a fourth derivative equation for x1 alone.

2) Write the differential equation as a matrix equation for the matrix (x1, x2) and find the eigenvectors and eigenvectors of the coefficient matrix.

 

[voko]

There are, obviously, many different ways to attack this problem. Using the center of mass and conservation of momentum, however, the system is reduced to a one-body problem.