Question

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (vertically). If the acceleration of the system is $g / 8$, then the ratio of the masses is :

• A. 8 : 1
• B. 9 : 7
• C. 4 : 3
• D. 5 : 3

Answer 1

Answer: (B)

As the string is inextensible, both masses have the same acceleration $a$. Also, the pulley is massless and frictionless, hence, the tension at both ends of the string is the same. Suppose the mass $m_{2}$ is greater than mass $m_{1},$ so, the heavier mass $m_{2}$ is accelerated downward and the lighter mass $m_{1}$ is accelerated upwards.

Therefore, by Newton's 2 nd law
$T-m_{1} g=m_{1} a$...(1)
$m_{2} g-T=m_{2} a$...(2)
After solving Eqs. (1) and (2)
$a=\frac{\left(m_{2}-m_{1}\right)}{\left(m_{1}+m_{2}\right)} \cdot g=\frac{g}{8}$(given)
so,$\frac{g}{8}=\frac{m_{2}\left(1-m_{1} / m_{2}\right)}{m_{2}\left(1+m_{1} / m_{2}\right)} \cdot g$
Let $\frac{m_{1}}{m_{2}}=x$
Thus Eq. (3) becomes
$\frac{1-x}{1+x}=\frac{1}{8}$
or $x=\frac{7}{9}$ or $\frac{m_{2}}{m_{1}}=\frac{9}{7}$
So, the ratio of the masses is 9: 7 .