Question

In the figure given below, with what acceleration does the block of mass $m$ will move? (Pulley and strings are massless and frictionless)

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

Answer 1

The correct option is C $\frac{2g}{3}$

Take $2m$ and $3m$ as a combined system (i.e., single block of $5m$ mass) as shown in the FBD.

For the single pulley system, $a=\left(\frac{m_1-m_2}{m_1+m_2}\right)g$
Where, $m_1=5m$ and $m_2=m$
$\Rightarrow a=\left(\frac{5m-m}{5m+m}\right)g=\frac{2g}{3}$

Alternate:
Let $a$ be the acceleration of whole system.
The FBDs of the individual masses are as shown

From the FBDs we have
$3mg-T_1=3ma$......(i)
$T_1+2mg-T=2ma$.....(ii)
So, from equation (i) & (ii), we have
$5mg-T=5ma$.....(iii)
Also, $T-mg=ma$....(iv)
Thus, from (iii) & (iv), we have
$4mg=6ma\Rightarrow a=\frac{4mg}{6m}=\frac{2g}{3}$