First slide

Rotational motion

Question

Each pulley in the figure has radius r and moment of inertia I. The acceleration of the block is

Moderate

A

$\frac{(M - m) g}{(M + m + \frac{2 I}{r^{2}})}$
B

$\frac{(M - m) g}{(M + m - \frac{2 I}{r^{2}})}$
C

$\frac{(M - m) g}{(M + m + \frac{I}{r^{2}})}$
D

$\frac{(M - m) g}{(M + m - \frac{I}{r^{2}})}$

Solution

$T_{1} - m g = m a$ ……..(i)
$\begin{aligned} r (T_{2} - T_{1}) = I α . . . . . . . . . (i i) \\ M g - T_{3} = M a . . . . . . . . . (i i i) \\ r (T_{3} - T_{2}) = I α . . . . . . . . . (i v) \end{aligned}$
and a = R $α$
From Eqs. (ii) and (iv), we get
$T_{3} - T_{1} = \frac{2 I α}{r} = \frac{2 I a}{R^{2}}$
From Eqs. (i) and (iii), we get
$\begin{array}{l} (M - m) g = (M + m) a + T_{3} - T_{1} \\ \Rightarrow (M - m) g = (M + m) a + \frac{2 I a}{r^{2}} \\ \Rightarrow a = \frac{(M - m) g}{(M + m + \frac{2 I}{r^{2}})} \end{array}$

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