A block of mass $m$ slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is: Given $m = 8 \mathrm {~kg} , \quad M = 16 \mathrm {~kg}$ Assume all the surfaces shown in the figure to be frictionless. (1) $\frac { 3 } { 5 } \mathrm {~g}$ (2) $\frac { 4 } { 3 } \mathrm {~g}$ (3) $\frac { 6 } { 5 } \mathrm {~g}$ (4) $\frac { 2 } { 3 } \mathrm {~g}$
A body of mass $m$ dropped from a height $h$ reaches the ground with a speed of $0.8 \sqrt { g h }$. The value of work done by the air-friction is: (1) $- 0.68 m g h$ (2) $m g h$ (3) 0.64 mgh (4) 1.64 mgh
Four particles each of mass $M$, move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is: (1) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R } (2 \sqrt { 2 } + 1)}$ (2) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R ( 2 \sqrt { 2 } + 1 ) } }$ (3) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R } (2 \sqrt { 2 } - 1)}$ (4) $\sqrt { \frac { G M } { R } }$
An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of $40,000 \mathrm {~kg}$ is moving with a speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If $90 \%$ of energy of the wagon is lost due to friction, the spring constant is $\_\_\_\_$ $\times 10 ^ { 5 } \mathrm {~N} \mathrm {~m} ^ { - 1 }$.
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 $x =$ $\_\_\_\_$.
A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in the vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of the rod when it passes through its lowest position is $\_\_\_\_$ $\mathrm { m } \mathrm { s } ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
The number of pairs $a , b$ of real numbers, such that whenever $\alpha$ is a root of the equation $x ^ { 2 } + a x + b = 0 , \quad \alpha ^ { 2 } - 2$ is also a root of this equation, is : (1) 6 (2) 8 (3) 4 (4) 2
Let $P _ { 1 } , \quad P _ { 2 } \ldots , \quad P _ { 15 }$ be 15 points on a circle. The number of distinct triangles formed by points $P _ { i } , \quad P _ { j } , \quad P _ { k }$ such that $i + j + k \neq 15$, is : (1) 455 (2) 419 (3) 12 (4) 443
Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 21 }$ be an A.P. such that $\sum _ { n = 1 } ^ { 20 } \frac { 1 } { a _ { n } a _ { n + 1 } } = \frac { 4 } { 9 }$. If the sum of this A.P. is 189 , then $\mathrm { a } _ { 6 } \mathrm { a } _ { 16 }$ is equal to : (1) 57 (2) 48 (3) 36 (4) 72
Consider the parabola with vertex $\left(\frac { 1 } { 2 } , \frac { 3 } { 4 }\right)$ and the directrix $y = \frac { 1 } { 2 }$. Let P be the point where the parabola meets the line $x = - \frac { 1 } { 2 }$. If the normal to the parabola at P intersects the parabola again at the point Q, then $(PQ) ^ { 2 }$ is equal to : (1) $\frac { 25 } { 2 }$ (2) $\frac { 75 } { 8 }$ (3) $\frac { 125 } { 16 }$ (4) $\frac { 15 } { 2 }$
Let $\theta$ be the acute angle between the tangents to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 } = 1$ and the circle $x^2 + y^2 = 3$ at their points of intersection. Then $\tan\theta$ is equal to: (1) $\frac{4}{\sqrt{3}}$ (2) $\frac{2}{\sqrt{3}}$ (3) $2$ (4) $\frac{5}{2\sqrt{3}}$