Find the acceleration of two blocks. Assume pulleys and strings are massless and frictionless. Also assume the string to be inxtensible.

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Solution

$Y + 2 X + Z = ℓ$ ...(i)
$Y + X + ℓ_{2} + X + ℓ_{2} + Z - ℓ_{1} = ℓ$ ...(ii)
Subtracting eq. $1^{s t}$ from eq. $2^{n d}$ , we get $2 ℓ_{2} = ℓ_{1}$
Differenttiting twice this equation $w . r . t$ time we get $∴ a_{1} = 2 a_{2}$
i.e. acceleration of block $m_{1}$ is twice the acceleration of block $m_{2}$
For block $m_{1}$
$T_{1} = m_{1} a_{1}$
For block $m_{2}$
$m_{2} g - T_{2} = m_{2} a_{2}$
$2 T_{1} = T_{2}$ [ $∵$ Pulley is massless]
Let $a_{2} = a ∵ T = 2 m_{1} a$
$∴ a_{1} = 2 a, m_{2} g - 2 T = m_{2} a$
$T_{1} = T$ Solving $a_{2} = \frac{m_{2} g}{m_{2} + 4 m_{1}}$
$∴ T_{2} = 2 T ∴ a_{1} = 2 a_{2} = \frac{2 m_{2} g}{m_{2} + 4 m_{1}}$

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