Question

A system consists of three masses $m_1$, $m_2$ and $m_3$ connected by a string passing over a pulley $P$. The mass $m_1$ hangs freely and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction = $\mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is (Assume $m_1 = m_2 = m_3 = m$)

• A. $\frac{g(1-g\mu)}{9}$
• B. $\frac{2g\mu}{3}$
• C. $\frac{g(1-2\mu)}{3}$
• D. $\frac{g(1-2\mu)}{2}$

Answer 1

Answer: (C)

$\frac{g(1-2\mu)}{3}$

Answer 2

The correct answer is C:$\frac{g(1-2\mu)}{3}$
Force of friction on mass $m_2=\mu m_2g$
Force of friction on mass $m_3=\mu m_3g$
Let a be common acceleration of the
system.
$\therefore a=\frac{m_1g-\mu m_2g-\mu m_3g}{m_1+m_2+m_3}$
Here, $m_1=m_2=m_3=m$
$\therefore a=\frac{mg-\mu mg-\mu mg}{m+m+m}=\frac{mg-2\mu mg}{3m}$
$=\frac{g(1-2\mu)}{3}$
Hence, the downward acceleration of
mass $m_1$ is $\frac{g(1-2\mu)}{3}$.