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Spring Force

In the arrang...

Question

In the arrangement shown in figure pulleys are smooth and massless. Threads are massless and inextensible. Find acceleration of mass $m_{1}$ .

\frac{[2 m_{1} m_{2} + m_{0} (m_{1} - m_{2}] g}{2 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}

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- \frac{[4 m_{1} m_{2} + m_{0} (m_{1} - m_{2}] g}{4 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}

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\frac{[4 m_{1} m_{2} - m_{0} (m_{1} - m_{2}] g}{4 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}

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Solution

The correct option is A $- \frac{[4 m_{1} m_{2} + m_{0} (m_{1} - m_{2}] g}{4 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}$
Here refer to the fig. for the scheme of the motion of the system
Let the tension in the lower string be $T$
Therefore tension in the upper string is $2 T$
Also let acceleration of both the mass $M_{1} {a n d M}_{2}$ w.r.t. the lower pulley be $a_{1}$
Also let the acceleration of lower pulley w.r.t. ground $= a$
So, acceleration of masses $M_{1} {a n d M}_{2}$ w.r.t. ground is $a_{1} - a a n d a_{1} + a r e s p e c t i v e l y$
So, the three equations of motion for masses $M_{0} {, M}_{1} {, M}_{2} r e s p e c t i v e l y a r e, 2 T = M_{0} a_{0} ⟶ (e q u a t i o n 1) T - M_{1} g = M_{1} (a_{1} - a_{0}) ⟶ (e q u a t i o n 2) M_{2} g - T = M_{2} (a_{1} + a_{0}) ⟶ (e q u a t i o n 3)$
Solving the three equations we have $a_{1} - a_{0} = \frac{- [4 m_{1} m_{2} + m_{0} (m_{1} - m_{2})] g}{{4 m}_{1} m_{2} + m_{0} (m_{1} {+ m}_{2})}$

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