- #1

1v1Dota2RightMeow

- 76

- 7

## Homework Statement

2 masses are connected by a spring. They are on a frictionless plane inclined relative to the horizontal by ##\alpha##. The masses are free to slide, rotate about their center of mass, and oscillate.

1. Find the Lagrangian as a sum of the Lagrangian for the COM motion and a Lagrangian for the relative motion.

2. From this equation, determine and solve the equations of motion for the COM and the relative motion.

3. Determine the constants of the motion

4. Determine the motions of the COM and the relative coordinate

## Homework Equations

##L_{total}=L_{COM}+L_{rel}##

##L=KE-PE##

##KE=(1/2)mv^2##

##PE=mgh##

##KE_{rot}=(1/2)I\omega ^2##

##PE_{spring}=(1/2)kx^2##

##I=mr^2## because we are dealing with point masses

##\omega = \frac{v}{r}##

## The Attempt at a Solution

1.

##L_{COM} = KE_{COM}-PE_{COM}##

##KE_{COM} = (1/2)Mv^2=(1/2)(\frac{m_1m_2}{m_1+m_2})v^2## , but what would ##v## be here? And am I correct in using the reduced mass?

##PE_{COM} = Mgh(t) = (\frac{m_1m_2}{m_1+m_2})gh(t)## , but how do I determine ##h(t)##?

##L_{rel}=KE_{rel}-PE_{rel}##

##KE_{rel}## is comprised of both rotational motion and oscillatory motion:

##KE_{rel} = (1/2)I_1\omega_1^2 + (1/2)I_2\omega_2^2 + (1/2)m_1v_1^2+(1/2)m_2v_2^2##

##PE_{rel} = (1/2)kd_1^2+(1/2)kd_2^2##, where ##d## is the distance past the equilibrium point.

I can't move on to the other parts without understanding this first part. Could someone please help me understand?