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Springs in Parallel and Series

Two blocks ar...

Question

Two blocks are placed on a smooth horizontal surface and connected by a string pulley arrangement as shown. If a force $F$ starts acting on block $m_{1}$ , then find the relation between acceleration of both masses and their values.

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Solution

From top to bottom, let us assume that the lengths of the strings are $l_{1}, l_{2}, l_{3}, l_{4}$
Therefore the total length is $l_{1} + l_{2} + l_{3} + l_{4} = l$ .
After movement, lengths be $l_{1} - y, l_{2} - y + x, l_{3} - y + x a n d l_{4} - y + x$
So the total length will be $l_{1} - y + l_{2} - y + x + l_{3} - y + x + l_{4} - y + x$
Since $l_{1} + l_{2} + l_{3} + l_{4} = l$ , we find that $y = 3 x / 4$
Therefore if $a_{1}$ and $a_{2}$ are the accelerations, then $a_{1}$ = $\frac{d^{2} x}{d t^{2}}$ and $a_{2} = \frac{d^{2} y}{d t^{2}}$
So the relationship is $\frac{a_{1}}{a_{2}} = \frac{3}{4}$
Therefore, since acceleration of COM = $\frac{F}{(m 1 + m 2)}$ ,

we can say that $a_{1} = \frac{F}{(m_{1} + (3 m_{2} / 4))}$ and $a_{2} = \frac{3 F}{(4 m_{1} + 3 m_{2})}$

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