A smooth block is released at rest on a $45^\circ$ incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is
(1) $\mu_\mathrm{k} = 1 - \frac{1}{\mathrm{n}^2}$
(2) $\mu_\mathrm{k} = \sqrt{1 - \frac{1}{\mathrm{n}^2}}$
(3) $\mu_s = 1 - \frac{1}{n^2}$
(4) $\mu_s = \sqrt{1 - \frac{1}{n^2}}$

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by
(1) $2\sin\phi$
(2) $2\cos\phi$
(3) $2\tan\phi$
(4) $\tan\phi$

A particle of mass 0.3 kg is subjected to a force $F = -kx$ with $k = 15 \mathrm{~N}/\mathrm{m}$. What will be its initial acceleration if it is released from a point 20 cm away from the origin?
(1) $3 \mathrm{~m}/\mathrm{s}^2$
(2) $15 \mathrm{~m}/\mathrm{s}^2$
(3) $5 \mathrm{~m}/\mathrm{s}^2$
(4) $10 \mathrm{~m}/\mathrm{s}^2$

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the insect and the surface is $1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ so that the insect does not slip is given by
(1) $\cot \alpha = 3$
(2) $\sec \alpha = 3$
(3) $\operatorname{cosec} \alpha = 3$
(4) $\cos \alpha = 3$

A block of weight $W$ rests on a horizontal floor with coefficient of static friction $\mu$. It is desired to make the block move by applying minimum amount of force. The angle $\theta$ from the horizontal at which the force should be applied and magnitude of the force $F$ are respectively.
(1) $\theta = \tan ^ { - 1 } ( \mu ) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$
(2) $\theta = \tan ^ { - 1 } \left( \frac { 1 } { \mu } \right) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$
(3) $\theta = 0 , F = \mu W$
(4) $\theta = \tan ^ { - 1 } \left( \frac { \mu } { 1 + \mu } \right) , F = \frac { \mu W } { 1 + \mu }$

A body starts from rest on a long inclined plane of slope $45^{\circ}$. The coefficient of friction between the body and the plane varies as $\mu = 0.3x$, where $x$ is distance travelled down the plane. The body will have maximum speed (for $g = 10\mathrm{~m/s^2}$) when $x =$
(1) 9.8 m
(2) 27 m
(3) 12 m
(4) 3.33 m

A bullet of mass 4 g is fired horizontally with a speed of $300 \mathrm{~m}/\mathrm{s}$ into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately?
(1) 0.19 m
(2) 0.379 m
(3) 0.569 m
(4) 0.758 m

A block of mass 5 kg is (i) pushed in case (A) and (ii) pulled in case (B), by a force $\mathrm { F } = 20 \mathrm {~N}$, making an angle of $30 ^ { \circ }$ with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is $\mu = 0.2$. The difference between the accelerations of the block, in case ( B ) and case ( A ) will be: ( $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ )
(1) $3.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(2) $0 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(3) $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
(4) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

Two blocks $A$ and $B$ of masses $m _ { A } = 1 \mathrm {~kg}$ and $m _ { B } = 3 \mathrm {~kg}$ are kept on the table as shown in figure. The coefficients of friction between $A$ and $B$ is 0.2 and between $B$ and the surface of the table is also 0.2 . The maximum force F that can be applied on B horizontally, so that the block A does not slide over the block B is : [Take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ ]
(1) 16 N
(2) 12 N
(3) 40 N
(4) 8 N

A block of mass 10 kg starts sliding on a surface with an initial velocity of $9.8 \mathrm {~ms} ^ { - 1 }$. The coefficient of friction between the surface and block is 0.5 . The distance covered by the block before coming to rest is :[use $\mathrm { g } = 9.8 \mathrm {~ms} ^ { - 2 }$ ]
(1) 9.8 m
(2) 4.9 m
(3) 12.5 m
(4) 19.6 m