Question

Two weights W and 2W are supported in a vertical plane by a string and pulleys arranged as shown. Find the magnitude of the additional weight Q applied on the left which will give a downward acceleration of 0.1g to the weight W. Neglect friction and inertia of the pulleys.

• A. W
• B. $\frac{W}{2}$
• C. $\frac{W}{4}$
• D. $\frac{W}{6}$

Answer 1

The correct option is D $\frac{W}{6}$

${\ddot{x}}_B=-\frac{1}{2}{\ddot{x}}_A$

$=-\frac{0.1}{2}g=0.05gm/s^2$ upwards

Kinetics

Consider the free body diagram of B. Since the friction is neglected, the tension in the string remains the same.



Writing the equation of motion of the pulley B

$\frac{2W}{g}{\ddot{x}}_B=2T-2W$

$T=W+\frac{W}{g}\times 0.05g=1.05W$

Consider the free body diagram of A.
Writing the equation of motion


$\left(\frac{W+Q}{g}\right){\ddot{x}}_A=(Q+W)-T$

$\left(\frac{W+Q}{g}\right)\times 0.1g=Q+W-1.05W$

$0.1W+0.1Q=Q-0.5W$

$0.9Q=0.15W$

$\therefore Q=\frac{W}{6}$