Simple Double Mass Spring System

[omegacore]

Homework Statement

A mass m1 is held a length L above a second mass, m2, which rests on a table. There is a massless spring connecting the masses with spring constant K. You throw mass m1 upwards with velocity v0, determine the equations of position for each mass. Standard idealized conditions apply (Gravity does matter).

Homework Equations

y = y1 - y2
Ycm = m1 y1 + m2 y2
M= m1+m2

Derivable
y1 = Ycm + m2/M (y)
y2 = Ycm - m1/M (y)

The Attempt at a Solution

My first step was to confirm y1 and y2 as they stand. After understanding their origin, I attempted to derive the kinematic expression for the center of mass' motion. I found

Ycm = -(1/2)gt² + (m1/M) v0 t + (m1/M) L

Next I attempted to find an expression for y (essentially the relative position vector for both masses). I did this by assuming I can fix myself in the intertial reference frame of m2 thus giving the expression for y as
y = y1 - 0

So then I said

m1 y1'' + k y1 = 0

Which gives me a solution of the form

y1 = y = A sin(wt) + B cos(wt)

With initial conditions

y = (v0/w) sin(wt) + L cos(wt)

When I use these two expressions to find the final equation my answer looks something like

y1 = -(1/2)gt² + (m1/M)v0 t + (m1/M)L + (m2/M)[ (v0/w)sin(wt) + L cos(wt) ]

The answer in the back of the book is


y1 = -(1/2)gt² + (m1/M)v0 t + L + (m2/M)[ (v0/w)sin(wt) ]

and

y2 = -(1/2)gt² + (m1/M)v0 t - (m1/M)[ (v0/w)sin(wt)]

I am very close. I am just missing something stupid. If someone could point me in the right direction, or just point out something very obvious which is escaping me I am sure I can be on my way again soon enough.

 

[Jhero]

Thank you for your help!




Thank you for your post. Your attempt at the solution looks good so far. However, there are a few things you can tweak to get to the correct answer.

First, for the expression for y, you can use y = y1 - y2 instead of y = y1 - 0. This will give you an expression for y2 as well, which you can then use in your final equations.

Also, for the final equations, you need to use the initial conditions for y1 and y2 at t=0 to get the correct expressions. This will give you the correct equations as given in the back of the book.

I hope this helps. Keep up the good work!