Question

A small cubical block of mass $1kg$ is placed inside a rough rectangular groove made in a circular rough table as shown in the figure. The coefficient of friction for all the rough surfaces is $\mu =0.5$. The table starts rotating clockwise with angular acceleration $1rad/s^2$ in a horizontal plane about its axis. Find the time (in $s$) after which the block will start moving with respect to the table. Assume the size of block slightly smaller than the width of groove.

Answer 1

Draw a FBD of given situation



Find the time after which the block will start moving.

Formula used: $\omega ={\omega }_0+\alpha t,f=\mu N$

With respect to table: $f_1=\mu N_1;f_2=\mu N_2$

Also at time $t$,

$\omega ={\omega }_0+\alpha t=t$

The block starts sliding if the resultant of $[-(m{\overrightarrow{a}}_c+m{\overrightarrow{a}}_t)+\overrightarrow{N_2}]$ becomes greater than the maximum total friction.

$\Rightarrow m{\omega }^{2}r\cos {37}^{\circ }-mar\cos {53}^{\circ }>\mu (m{\omega }^{2}r\sin {37}^{\circ }+mar\sin {53}^{\circ })+\mu mg$

$\Rightarrow \frac{4t^2}{5}-\frac{3}{5}>\frac{1}{2}[\frac{3}{5}{t}^2+\frac{4}{5}+\frac{g}{r}]$

$\Rightarrow t^2>4\Rightarrow t>2s$

Final Answer: 2 s