$A$ box is lying on an inclined plane. What is the coefficient of static friction if the box starts sliding when the angle of inclination is $60^\circ$?

$A$ cone of radius $r$ and height $h$ rests on a rough horizontal surface,the coefficient of friction between the cone and the surface being $\mu$. $A$ gradually increasing horizontal force $F$ is applied to the vertex of the cone. The largest value of $\mu$ for which the cone will slide before it topples is

When a body slides down from rest along a smooth inclined plane making an angle of $30^{\circ}$ with the horizontal,it takes time $T$. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance,it takes time $\alpha T$,where $\alpha$ is a constant greater than $1$. The coefficient of friction between the body and the rough plane is $\frac{1}{\sqrt{x}}\left(\frac{\alpha^{2}-1}{\alpha^{2}}\right)$ where $x = .....$.

In the figure,two blocks $M$ and $m$ are tied together with an inextensible and light string. The mass $M$ is placed on a rough horizontal surface with a coefficient of friction $\mu$,and the mass $m$ is hanging vertically. The pulley is frictionless. Choose the correct statement$(s)$.

$A$ block is released from rest on a $45^\circ$ smooth incline and slides a distance '$d$'. The time taken to slide the same distance '$d$' on a rough incline is '$n$' times the time taken on the smooth incline. The coefficient of kinetic friction is

$A$ particle of mass $m$ slides from rest down a plane inclined at $30^{\circ}$ to the horizontal. The force of resistance acting on the particle during motion is $ms^2$,where $s$ is the displacement of the particle from its initial position. The velocity (in $m/s$) of the particle when $s = 1\,m$ is $v$. The value of $\frac{3v^2}{14}$ is: