First slide

Rotational motion

Question

Each pulley in 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

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

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