Question

In the figure shown, all pulleys are massless and the strings are light and inextensible. What is the acceleration of the pulley $P_4$? (All surfaces are smooth and take $g=10m/s^2$.)

• A. $2.25{\text{m/s}}^2$ towards left
• B. $2.25{\text{m/s}}^2$ towards right
• C. $9{\text{m/s}}^2$ towards left
• D. $9{\text{m/s}}^2$ towards right

Answer 1

The correct option is B $2.25{\text{m/s}}^2$ towards right

The given system can be reframed as shown

FBDs of the pulleys and the block

For $P_7:80-2T=m{a}_7\Rightarrow T=40\text{N}$
As pulley is massless, $m=0$
For the block: The acceleration of the block $a$ is given by
$10g\sin 30°-\frac{40}{8}=10a$
$\Rightarrow a=4.5{\text{m/s}}^2$
For constrained condition, let pulley $P_4$ move to the left with $a_p$
${\stackrel{..}{L}}_{11}+{\stackrel{..}{L}}_{10}+{\stackrel{..}{L}}_9+{\stackrel{..}{L}}_8+{\stackrel{..}{L}}_7=0$
$\Rightarrow a+0+0+a_p+a_p=0$
$\Rightarrow a_p=-\frac{a}{2}=-\frac{4.5}{2}=-2.25m/s^2$
Thus, $a_p=-2.25{\text{m/s}}^2$ to left
or $2.25{\text{m/s}}^2$ to right.