$A$ body is moving down a long inclined plane of slope $37^{\circ}$. The coefficient of friction between the body and the plane varies as $\mu = 0.3x$,where $x$ is the distance travelled down the plane. The body will have maximum speed at:

Define the condition for a vehicle to be parked on a banked slope without slipping.

$A$ block of mass $5\,kg$ and surface area $2\,m^2$ just begins to slide down an inclined plane when the angle of inclination is $30^{\circ}$. Keeping the mass the same,the surface area of the block is doubled. The angle at which this starts sliding down is:

$A$ block of mass $5 kg$ is at rest on a rough inclined surface. If the angle of inclination of the plane is $60^{\circ}$,then the total force applied by the surface on the block is .......... $N$. (Take $g = 10 m/s^2$)

$A$ rectangular box lies on a rough inclined surface. The coefficient of friction between the surface and the box is $\mu$. Let the mass of the box be $m$.
$(a)$ At what angle of inclination $\theta$ of the plane to the horizontal will the box just start to slide down the plane?
$(b)$ What is the force acting on the box down the plane,if the angle of inclination of the plane is increased to $\alpha > \theta$?
$(c)$ What is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed?
$(d)$ What is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration $a$?

$A$ 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: