First slide

Rotational motion

Question

A cord is wrapped on a pulley (disc) of mass M and radius R as shown in the figure. To one end of the cord, a block of mass M is connected as shown and to other end
in (a) a force of 2 Mg and in (b) a block of mass 2M. Let angular acceleration of the disc in (a) and (b) is $α_{A}$ and $α_{B}$ , respectively, then (cord is not slipping on the pulley).

Moderate

A

$α_{A} = α_{B}$
B

$α_{A} > α_{B}$
C

$α_{A} < α_{B}$
D

none of these

Solution

$\begin{array}{l} In the figure, (T_{2} - T_{1}) R = \frac{M R^{2}}{2} α_{A} \\ \begin{aligned} T_{1} - M g = M a_{A} \\ T_{2} = 2 M g \\ a_{A} = R α_{A} \\ α_{A} = \frac{2 g}{3 R} \end{aligned} \end{array}$
From the figure,
$\begin{array}{l} (T_{2} - T_{1}) \times R = \frac{M R^{2}}{2} α_{B} \\ T_{1} - M g = M a_{B} \\ 2 M g - T_{2} = 2 M a_{B} \\ a_{B} = R α_{B} \\ α_{B} = \frac{2 g}{7 R} \\ So, α_{A} > α_{B} \end{array}$

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