Question

In the arrangement shown in the figure, the rod and the sleeve have friction coefficient of $\frac{1}{3}$. The mass of the hanging block is $2kg$. The sleeve is just on the verge of sliding down. What is the mass (in $kg$) of the sleeve.

Answer 1

Let $m_b$ and $m$ be the mass of the block and sleeve respectively and $T$ be the tension in the string.

From the FBD of the block,


$\begin{array}{}T=m_{b}g=2g& \text{(1)}\end{array}$
FBD of sleeve,

$\begin{array}{}T\sin {37}^{\circ }=N& \text{(2)}\end{array}$
$T\cos {37}^{\circ }+\mu N=mg\text{Using}\left(1\right)\text{and}\left(2\right),\Rightarrow 2g(\frac{4}{5}+\frac{1}{3}\times \frac{3}{5})=mg2\times \frac{5}{5}=m$
$\Rightarrow m=2kg$