# Relative acceleration with pullys

I have tried this question every way I can think but in the equation for particle $L$ $g$ cancels every time. Could someone show me how to do it correctly or tell me what I am doing wrong. Thanks,

The Question:

A string with negligible mass passes over a smooth pulley V with a particle A of mass $18kg$ on one end of the string and a pulley ($J$) of negligible mass on the the other end

Another string with negligible mass passes over pulley $J$ and has a particle $K$ of mass $12kg$ on one end and a particle $L$ of mass $9kg$ on the other end.

Show the common acceleration of $A$ and $J$ then show the relative acceleration of $K$ and $L$ to $J$.

So far I have worked out that the tension in the top string is equal to twice the tension in the bottom string. $T-2S=0a$ $$T=2S$$

I then plug this into $18g-T=18a$ to get $18g-2S=18a$

From that equation I get $S=9g-9a$

After that I plug the value of $S$ into the equations for $K$ and $L$ then in the equation for $L$ $g$ is canceled out and I am stuck.

• You should post what you have tried so far. It might be that you are making a small mistake. And it generally helps answerers to decide how much detail to include in an answer, so that you can understand it. – udiboy1209 Aug 17 '13 at 17:45
• Maybe a sketch, and your work so far are in order. – ja72 Aug 17 '13 at 19:27

The accelerations of K and L will be different from one another and also from the particle A. To solve this consider $a_p$ to be the acceleration of the pulley J and $a_r$ be the relative accelerations of the particles K and L relative to the frame of reference of the pulley J. Then the net acceleration of the particle K will be $a_p-a_r$ and that of particle L will be $a_p+a_l$. Now apply the equations for the particles K and L and the acceleration term wont cancel out. ( you can interchange the accelerations of both the particles and the only change it will bring is the sign of the answer and that will tell you the actual direction of the particles K and L)
for k $$S-12g=12(a_p-a_r)$$ and for L $$S-9g=9(a_p+a_r)$$. solve this and you will get your answer.