Velocity and Acceleration of the Center of Mass?

[Scarlitt14]

Homework Statement

Given: g=9.8 m/s^s
Given: A uniform flexible chain whose mass is 7 kg and length is 5 m.
Given: A small frictionless pulley whose circumference is negligible compared to the length of the chain.

Problem: Initially the chain is hung over the pulley with nearly equal lengths of both side but just unequal enough so that the unstable equilibrium condition will let the chain start to move. After some time, the longer end of the chain is a distance l=3.8m down from the pulley's axle. Find the acceleration aof the chain when the chain is at this position. Find the velocity v of the chain when l=3.8 m.

Homework Equations

y_cm=\frac{y_{1}m_{1}+y_{2}m_{2}}{m_{total}}

\sum\vec{F}=m\vec{a}

\vec{a}_cm=\frac{1}{M}\summ_i\vec{a}_i=\sum\vec{F}

\vec{v}_cm=\frac{1}{M}\summ_i\vec{v}_i=\sum\vec{F}

The Attempt at a Solution

I've attempted this problem in a half a dozen different ways. I used the above equations, I realigned my axes to the pulley and to the bottom of the system, I drew free body diagrams for each side separately and the system as a whole, I even used kinematics to find the time to travel the given distance (Why? I'm not quite sure!) I don't know what other methods to use. I'm just hoping that its not something silly like I miscalculated my center of mass!

Any help is greatly appreciated!

 

[rl.bhat]

At 3.8 m, take m1 as (M/L)*3.8, where M is the total mass and L is the total length. Similarly find m2. Taking same tension on either side of the chain, find the acceleration. It is the same on either side.

 

[Scarlitt14]

I used the mass density (M/L) in both my center of mass equations and the sum of forces. I'm able to find the acceleration for each piece of the chain, however, I cannot for the life of me figure out how to find the acceleration of the chain as a whole (acceleration of the center of mass I think). Same thing with velocity, I can find the individual velocities but not for the center of mass.

 

[rl.bhat]

In the rigid body you can find the center of mass. In the chain mass distribution continuously changes during its motion.