Question

A light thread is wound on a disk of mass $m$ and other end of thread is connected to a block of mass $m$, which is placed on a rough ground as shown in diagram. Find the minimum value of coefficient of friction for which block remain at rest.

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{5}$

Answer 1

The correct option is A $\frac{1}{3}$

The FBD of two would be like as shown in figure.



For disc: $mg-T=ma$,
No slipping condition: $a_{12}=0\Rightarrow r\alpha -a=0$
$Tr=\frac{m{r}^2}{2}\times \frac{a}{r}$
Solving above equations, we get $a=\frac{2g}{3}$ and
For block to remain at rest, for minimum $\mu$ case,
$T=f$ and $f\le \mu mg$
$T=\frac{mg}{3}=\mu mg$
$\mu =\frac{1}{3}$