An insect is at the bottom of a hemispherical ditch of radius $R = 1\, m$. It crawls up the ditch but starts slipping after it is at height $h$ from the bottom. If the coefficient of friction between the ground and the insect is $\mu = 0.75$,then $h$ is $.......\, m$. $(g = 10\, m s^{-2})$

An engine of mass $1$ metric ton is ascending an inclined plane,at an angle $\theta = \tan^{-1}(1/2)$ with the horizontal,with a speed of $36 \; km/h$. If the coefficient of friction of the surface is $1/\sqrt{3}$,then the power developed by the engine is:

$A$ uniform chain of total length $L$ is at rest,partially on an incline of angle $\theta = 30^{\circ}$ and partially hanging vertically. The coefficient of friction between the chain and the incline is $\mu = \frac{1}{2\sqrt{3}}$. Find the ratio $\frac{L_{\max}}{L_{\min}}$,where $L_{\max}$ is the maximum length of the chain on the incline and $L_{\min}$ is the minimum length of the chain on the incline such that the chain remains at rest.

$A$ block of mass $m$ is placed on a surface with a vertical cross-section given by $y = \frac{x^3}{6}$. If the coefficient of friction is $0.5$,the maximum height above the ground at which the block can be placed without slipping is:

$A$ block is placed on a rough inclined plane of inclination $\theta = 30^o$. If the force required to drag it up along the plane is to be smaller than the force required to lift it vertically,the coefficient of friction $\mu$ should be less than:

$A$ block of mass $5 \ kg$ is moving on an inclined plane which makes an angle of $30^{\circ}$ with the horizontal. The coefficient of friction between the block and the inclined plane surface is $\frac{\sqrt{3}}{2}$. The force to be applied on the block so that the block will move down without acceleration is . . . . . . $N$.