Q. A rocket is fired vertically from the earth with an acceleration of $2g$, where $g$ is the gravitational acceleration. On an inclined plane inside the rocket, making an angle $\theta$ with the horizontal, a point object of mass $m$ is kept. The minimum coefficient of friction $\mu_{min}$ between the mass and the inclined surface such that the mass does not move is :
Solution:
Free body diagram is shown in the figure.
Psuedo force on the object is $2 m g$ in the downward direction as seen from the rocket frame of reference.
Normal force in the direction perpendicular to the inclined plane.
$ N=3 m g \cos (\theta)
$ Maximum frictional force in the direction along the plane,
$ F_{r}=\mu_{m} N $
$\therefore F_{r}=3 \mu_{m} m g \cos (\theta) $
This should balance the force on the block along the downward direction of plane.
$ 3 m g \sin (\theta)=3 \mu_{m} m g \cos (\theta) $
$\mu_{m}=\tan (\theta)$
