163. In the figure, a weight is thrown upward from the bottom of an inclined surface with initial velocity $V_0$, tangent to the surface. The weight goes up and returns to the starting point. If the friction force is $\frac{2}{10}$ of the weight, the time to go up equals the time to come down by how much? ($g = 10\,\frac{m}{s^2}$)
[Figure: inclined surface at $30^\circ$ with initial velocity $V_0$ along the surface]
(1) $\sqrt{\dfrac{4}{3}}$ (2) $\sqrt{\dfrac{3}{7}}$ (3) $\dfrac{3}{5}$ (4) $\dfrac{5}{3}$

A small block slides down from the top of hemisphere of radius $R = 3 \mathrm {~m}$ as shown in the figure. The height $h$ at which the block will lose contact with the surface of the sphere is $\_\_\_\_$ m. (Assume there is no friction between the block and the hemisphere)

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 the 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 co-efficient of friction between the body and the rough plane is $\frac { 1 } { \sqrt { x } } \frac { \alpha ^ { 2 } - 1 } { \alpha ^ { 2 } }$ where $x =$ $\_\_\_\_$.

Q3. A given object takes n times the time to slide down $45 ^ { \circ }$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45 ^ { \circ }$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :
(1) $\sqrt { 1 - \frac { 1 } { n ^ { 2 } } }$
(2) $1 - n ^ { 2 }$
(3) $1 - \frac { 1 } { n ^ { 2 } }$
(4) $\sqrt { 1 - n ^ { 2 } }$

Q4. A car of 800 kg is taking turn on a banked road of radius 300 m and angle of banking $30 ^ { \circ }$. If coefficient of static friction is 0.2 then the maximum speed with which car can negotiate the turn safely: $\left( \mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } , \sqrt { 3 } = 1.73 \right)$
(1) $264 \mathrm {~m} / \mathrm { s }$
(2) $51.4 \mathrm {~m} / \mathrm { s }$
(3) $70.4 \mathrm {~m} / \mathrm { s }$
(4) $102.8 \mathrm {~m} / \mathrm { s }$

Q23. A circular disc reaches from top to bottom of an inclined plane of length $l$. When it slips down the plane, if takes $t \mathrm {~s}$. When it rolls down the plane then it takes $\left( \frac { \alpha } { 2 } \right) ^ { 1 / 2 } t \mathrm {~s}$, where $\alpha$ is $\_\_\_\_$