First slide

Rotational motion

Question

A cord is wrapped on a pulley (disk) of mass M and radius R as shown in 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 disk 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

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

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