Question

A block of mass m is kept on rough horizontal turn table at a distance r from the centre of table. The coefficient of friction between turn table and block is $\mu$. Now the turn-table starts rotating with uniform angular acceleration $\alpha$.
a. Find the time after which slipping occurs between block and turn-table.
b. Find the angular made by friction force with velocity at the point of slipping.

Answer 1

Tangential acceleration of table, $a_1=\alpha r$
velocity after time t,

$v=u+a_{1}t$

$v=0+\alpha rt=\alpha rt$
Centripetal acceleration, $a_c=\frac{v^2}{r}={\alpha }^{2}r{t}^2$
Net acceleration, $a_{net}=\sqrt{a_t^2+a_c^2}=\sqrt{{\alpha }^2{r}^2+{\alpha }^4{r}^2{t}^2}$

For non-slipping condition
$\mu mg\ge m{a}_{net}=m\sqrt{{\alpha }^2{r}^2+{\alpha }^4{r}^2{t}^4}$
(a) $t={\left(\frac{{\mu }^2{g}^2-{\alpha }^2{r}^2}{{\alpha }^2{r}^2}\right)}^{1/4}={[\left(\frac{\mu g}{{\alpha }^{2}r}\right)-{\left(\frac{1}{\alpha }\right)}^2]}^{1/4}$
(b) velocity is in tangential direction but friction force is in radial direction

Direction of velocity is in the direction of acceleration.

$tan\theta =\frac{a_c}{a_r}=\frac{{\alpha }^{2}r{t}^2}{\alpha r}$
or $\theta =ta{n}^{-1}\left(\alpha t^2\right)$