Question

Two blocks of masses $m_1=3m$ and $m_2=m$ are attached to the ends of a string which passes over a frictionless fixed pulley (which is a uniform disc of mass $M=2m$ and radius R) as shown in Fig. The masses are then released. The acceleration of the system is

• A. 2g
• B. $\frac{2g}{3}$
• C. $\frac{2g}{5}$
• D. $\frac{2g}{7}$

Answer 1

The correct option is C

$\frac{2g}{5}$

Refer to Fig. Since the pulley has a finite mass, two tensions $T_{1}and{T}_2$ will not be equal. If a is the acceleration of the system, the equations of motion of masses are

$m_{1}g-T_1=m_{1}a\Rightarrow 3mg-T_1=3ma$ (1) and

$(T_2-m_{2}g=m_{2}a\Rightarrow T_2-mg=ma$ (2)

The resultant tension $(T_1-T_2)$ exerts a torque on the pulley which is given by

$t=(T_1-T_2)R$ (3)

Also, $t=I\alpha$ where $I=\frac{1}{2}M{R}^{2}and\alpha$ is the angular acceleration which is given by $\alpha =\frac{a}{R}$

$\therefore t=\frac{1}{2}M{R}^2\times \frac{a}{R}=\frac{1}{2}MRa=mRa$ (4)

$(\because M=2m)$

From Eqs. (3) and (4) we get

$T_2-T_2=ma$ (5)

Using Eqs. (1) and (2) and (5) and solving for a we get $a=\frac{2g}{5}$.