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For a rocket moving in free space, the fraction of mass to be disposed of Off to attain a speed equal to two times the exhaust speed is given by (given ${\mathrm{e}}^2$ = 7.4)
1. 0.40
2.  0.37
3.  0.50
4.  0.86

While walking on ice one should take small steps to avoid slipping. This is because smaller steps ensure
1. Larger friction
2. Smaller friction
3. Larger normal force
4. Smaller normal force

A block is kept on a rough inclined plane with angle of inclination $\theta$. The graph of net reaction (R) versus $\theta$ is:
1. 
2. 
3. 
 
(4) None of these

A block is placed on a rough horizontal plane. A time dependent horizontal force F=kt acts on the block. The acceleration time graph of the block is :
1. 
2. 
3. 
4. 

The forces required to just move a body up an inclined plane is double the force required to just prevent it from sliding down. If $\mathrm{\varphi }$ is angle of friction and $\mathrm{\theta }$ is the angle which the plane makes with horizontal, then
1. $\tan \theta =\tan \varphi$
2. $\tan \theta =2\tan \varphi$
3. $\tan \theta =3\tan \varphi$
4. $\tan \varphi =3\tan \theta$

The engine of a car produces an acceleration of 4 m/s² in the car. If this car pulls another car of the same mass, what will be the acceleration produced?
(1) 8 m/s²
(2) 2 m/s²
(3) 4 m/s²
(4) $\frac{1}{2}m/s^2$   

Find the maximum velocity for skidding for a car moved on a circular track of radius 100 m. The coefficient of friction between the road and tyre is 0.2 
(1) 0.14 m/s
(2) 140 m/s
(3) 1.4 km/s
(4) 14 m/s

A machine gun is mounted on a 2000 kg car on a horizontal frictionless surface. At some instant the gun fires bullets of mass 10 gm with a velocity of 500 m/sec with respect to the car. The number of bullets fired per second is ten. The average thrust on the system is 
(1) 550 N
(2) 50 N
(3) 250 N
(4) 250 dyne

The ratio of the weight of a man in a stationary lift and when it is moving downward with uniform acceleration ‘a’ is 3 : 2. The value of ‘a’ is (g-Acceleration due to gravity of the earth) 
1. $\frac{3}{2}g$
2. $\frac{g}{3}$
3. $\frac{2}{3}g$
4. g

If force on a rocket having exhaust velocity of 300 m/sec is 210 N, then rate of combustion of the fuel is 
(1) 0.7 kg/s
(2) 1.4 kg/s
(3) 0.07 kg/s
(4) 10.7 kg/s

A 5000 kg rocket is set for vertical firing. The exhaust speed is 800 ms^–1. To give an initial upward acceleration of 20 ms^–2, the amount of gas ejected per second to supply the needed thrust will be (g = 10 ms^–2) 
(1) 127.5 kg s^–1
(2) 187.5 kg s^–1
(3) 185.5 kg s^–1
(4) 137.5 kg s^–1

Two balls of masses m₁ and m₂ are separated from each other by a powder charge placed between them. The whole system is at rest on the ground. Suddenly the powder charge explodes and masses are pushed apart. The mass m₁ travels a distance s₁ and stops. If the coefficients of friction between the balls and ground are same, the mass m₂ stops after travelling the distance 
(1) $s_2=\frac{m_1}{m_2}{s}_1$
(2) $s_2=\frac{m_2}{m_1}{s}_1$
(3) $s_2=\frac{m_1^2}{m_2^2}{s}_1$
(4) $s_2=\frac{m_2^2}{m_1^2}{s}_1$

A cricket ball of mass 250 g collides with a bat with velocity 10 m/s and returns with the same velocity within 0.01 second. The force acted on bat is 
(1) 25 N
(2) 50 N
(3) 250 N
(4) 500 N

N bullets each of mass m kg are fired with a velocity v ms^–1 at the rate of n bullets per second upon a wall. The reaction offered by the wall to the bullets is given by:
(1) nmv
(2) $\frac{Nmv}{n}$
(3) $n\frac{Nm}{v}$
(4) $n\frac{Nv}{m}$

A particle moves in the XY-plane under the action of a force F such that the components of its linear momentum p at any time t are \(p_x = 2 \cos t\),  \(p_y = 2 \sin t\) . The angle between F and p at time t will be:

A satellite in force-free space sweeps stationary interplanetary dust at a rate $dM/dt=\alpha v$ where M is the mass, v is the velocity of the satellite and $\alpha$ is a negative constant. What is the deacceleration of the satellite?
(1) $-2\alpha v^2/M$
(2) $-\alpha v^2/M$
(3) $+\alpha v^2/M$
(4) $-\alpha v^2$

A rocket has a mass of 100 kg. 90% of this is fuel. It ejects fuel vapours at the rate of 1 kg/sec with a velocity of 500 m/sec relative to the rocket. It is supposed that the rocket is outside the gravitational field. The initial upthrust on the rocket when it just starts moving upwards is:
(1) Zero
(2) 500 N
(3) 1000 N
(4) 2000 N

Rocket engines lift a rocket from the earth surface because hot gases with high velocity:
(1) push against the earth.
(2) push against the air.
(3) react against the rocket and push it up.
(4) heat up the air which lifts the rocket.

Two masses m₁ and m₂ (m₁ > m₂) are connected by massless flexible and inextensible string passed over massless and frictionless pulley. The acceleration of centre of mass is 
(1) ${\left(\frac{{\displaystyle m_1-m_2}}{{\displaystyle m_1+m_2}}\right)}^{2}g$
(2) $\frac{m_1-m_2}{m_1+m_2}g$
(3) $\frac{m_1+m_2}{m_1-m_2}g$
(4) Zero

The mass of a body measured by a physical balance in a lift at rest is found to be m. If the lift is going up with an acceleration a, its mass will be measured as:
(1) $m\left(1-\frac{{\displaystyle a}}{{\displaystyle g}}\right)$
(2) $m\left(1+\frac{{\displaystyle a}}{{\displaystyle g}}\right)$
(3) m
(4) Zero

Three weights W, 2W and 3W are connected to identical springs suspended from a rigid horizontal rod. The assembly of the rod and the weights fall freely. The positions of the weights from the rod are such that 
(1) 3W will be farthest
(2) W will be farthest
(3) All will be at the same distance
(4) 2W will be farthest

A false balance has equal arms. An object weighs X when placed in one pan and Y when placed in other pan, then the weight W of the object is equal to 
(1) $\sqrt{XY}$
(2) $\frac{X+Y}{2}$
(3) $\frac{X^2+Y^2}{2}$
(4) $\frac{2}{\sqrt{X^2+Y^2}}$ 

In the arrangement shown in the figure, the ends P and Q of an unstretchable string move downwards with uniform speed U. Pulleys A and B are fixed. Mass M moves upwards with a speed 

1. $2U \cos \theta$
2. $U \cos \theta$
3. $\frac{2U}{\cos \theta }$
4. $\frac{U}{\cos \theta }$

A string of negligible mass going over a clamped pulley of mass m supports a block of mass M as shown in the figure. The force on the pulley by the clamp is given by:
                 
1. $\sqrt{2}Mg$
2. $\sqrt{2}mg$
3. $\sqrt{{(M+m)}^2+m^2}g$
4.$\sqrt{{(M+m)}^2+M^2}g$

A pulley fixed to the ceilling carries a string with blocks of mass m and 3 m attached to its ends. The masses of string and pulley are negligible. When the system is released, its centre of mass moves with what acceleration 
(1) 0
(2) g/4
(3) g/2
(4) –g/2

A solid sphere of mass 2 kg is resting inside a cube as shown in the figure. The cube is moving with a velocity $v=(5t\hat{i}+2t\hat{j})m/s$. Here t is the time in second. All surface are smooth. The sphere is at rest with respect to the cube. What is the total force exerted by the sphere on the cube? (Take g = 10 m/s²)

1. $\sqrt{29}N$
2. 29 N
3. 26 N
4. $\sqrt{89}N$

A block B is placed on top of block A. The mass of block B is less than the mass of block A. Friction exists between the blocks, whereas the ground on which block A is placed is assumed to be smooth. A horizontal force F, increasing linearly with time begins to act on B. The acceleration a_A and a_B of blocks A and B respectively are plotted against t. The correctly plotted graph is:
1.              
2.
3.              
4.

The force-time (F – t) curve of a particle executing linear motion is as shown in the figure. The momentum acquired by the particle in time interval from zero to 8 second will be 

(1) – 2 N-s
(2) + 4 N-s
(3) 6 N-s
(4) Zero

A body of 2 kg has an initial speed 5ms^–1. A force acts on it for some time in the direction of motion. The force time graph is shown in figure. The final speed of the body.

(1) 9.25 ms^–1
(2) 5 ms^–1
(3) 14.25 ms^–1
(4) 4.25 ms^–1

A particle of mass m, initially at rest, is acted upon by a variable force F for a brief interval of time T. It begins to move with a velocity u after the force stops acting. F is shown in the graph as a function of time. The curve is an ellipse.

(1) $u=\frac{\pi F_0^2}{2m}$
(2) $u=\frac{\pi T^2}{8m}$
(3) $u=\frac{\pi F_{0}T}{4m}$
(4) $u=\frac{F_{0}T}{2m}$

The variation of momentum with the time of one of the bodies in a two-body collision is shown in fig. The instantaneous force is the maximum corresponding to the point:
   
1. P
2. Q
3. R
4. S

The masses of 10 kg and 20 kg respectively are connected by a massless spring as shown in figure. A force of 200 N acts on the 20 kg mass. At the instant shown, the 10 kg mass has acceleration 12 m/sec². What is the acceleration of 20 kg mass?

(1) 12 m/sec²
(2) 4 m/sec²
(3) 10 m/sec²
(4) Zero

A uniform rope of length l lies on a table. If the coefficient of friction is μ, then the maximum length l₁ of the part of this rope which can overhang from the edge of the table without sliding down is 
(1) $\frac{l}{\mu }$
(2) $\frac{l}{\mu +l}$
(3) $\frac{\mu l}{1+\mu }$
(4) $\frac{\mu l}{\mu -1}$

A heavy uniform chain lies on a horizontal table-top. If the coefficient of friction between the chain and table surface is 0.25, then the maximum fraction of length of the chain, that can hang over one edge of the table is 
(1) 20%
(2) 25%
(3) 35%
(4) 15%

The blocks A and B are arranged as shown in the figure. The pulley is frictionless. The mass of A is 10 kg. The coefficient of friction of A with the horizontal surface is 0.20. The minimum mass of B to start the motion will be 

(1) 2 kg
(2) 0.2 kg
(3) 5 kg
(4) 10 kg

When two surfaces are coated with a lubricant, then they 
(1) Stick to each other
(2) Slide upon each other
(3) Roll upon each other
(4) None of these

A 20 kg block is initially at rest on a rough horizontal surface. A horizontal force of 75 N is required to set the block in motion. After it is in motion, a horizontal force of 60 N is required to keep the block moving with constant speed. The coefficient of static friction is 
(1) 0.38
(2) 0.44
(3) 0.52
(4) 0.60

A block A with mass 100 kg is resting on another block B of mass 200 kg. As shown in figure a horizontal rope tied to a wall holds it. The coefficient of friction between A and B is 0.2 while coefficient of friction between B and the ground is 0.3. The minimum required force F to start moving B will be 

(1) 900 N
(2) 100 N
(3) 1100 N
(4) 1200 N

Two carts of masses 200 kg and 300 kg on horizontal rails are pushed apart. Suppose the coefficient of friction between the carts and the rails are same. If the 200 kg cart travels a distance of 36 m and stops, then the distance travelled by the cart weighing 300 kg is 

(1) 32 m
(2) 24 m
(3) 16 m
(4) 12 m

A block of mass 50 kg can slide on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.6. The least force of pull acting at an angle of 30° to the upward drawn vertical which causes the block to just slide is 
(1) 29.43 N
(2) 219.6 N
(3) 21.96 N
(4) 294.3 N

On the horizontal surface of a truck (μ = 0.6), a block of mass 1 kg is placed. If the truck is accelerating at the rate of 5m/sec² then frictional force on the block will be 
(1) 5 N
(2) 6 N
(3) 5.88 N
(4) 8 N

A block of mass M = 5 kg is resting on a rough horizontal surface for which the coefficient of friction is 0.2. When a force F = 40 N is applied, the acceleration of the block will be (g = 10 m/s²) 

(1) 5.73 m/sec²
(2) 8.0 m/sec²
(3) 3.17 m/sec²
(4) 10.0 m/$se{c}^2$

A given object takes n times as much time to slide down a 45° rough incline as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is given by
(1) $\left(1-\frac{1}{n^2}\right)$
(2) $\frac{1}{1-n^2}$
(3) $\sqrt{\left(1-\frac{1}{n^2}\right)}$
(4) $\sqrt{\frac{1}{1-n^2}}$

Starting from rest, a body slides down a 45° inclined plane in twice the time it takes to slide down the same distance in the absence of friction. The coefficient of friction between the body and the inclined plane is 
(1) 0.33
(2) 0.25
(3) 0.75
(4) 0.80

A force of 750 N is applied to a block of mass 102 kg to prevent it from sliding on a plane with an inclination angle 30° with the horizontal. If the coefficients of static friction and kinetic friction between the block and the plane are 0.4 and 0.3 respectively, then the frictional force acting on the block is 
(1) 750 N
(2) 500 N
(3) 345 N
(4) 250 N

A body takes just twice the time as long to slide down a plane inclined at 30^o to the horizontal as if the plane were frictionless. The coefficient of friction between the body and the plane is
(1) $\frac{\sqrt{3}}{4}$
(2) $\sqrt{3}$
(3) $\frac{4}{3}$
(4) $\frac{3}{4}$

A body takes time t to reach the bottom of an inclined plane of angle θ with the horizontal. If the plane is made rough, time taken now is 2t. The coefficient of friction of the rough surface is
(1) $\frac{3}{4}\tan \theta$
(2) $\frac{2}{3}\tan \theta$
(3) $\frac{1}{4}\tan \theta$
(4) $\frac{1}{2}\tan \theta$

A block of mass 0.1 kg is held against a wall by applying a horizontal force of 5 N on the block. If the coefficient of friction between the block and the wall is 0.5, the magnitude of the frictional force acting on the block is 
(1) 2.5 N
(2) 0.98 N
(3) 4.9 N
(4) 0.49 N

What is the maximum value of the force \(F\) such that the block shown in the arrangement, does not move?
 
1. \(20~\)N
2. \(10~\)N
3. \(12~\)N
4. \(15~\)N

A 40 kg slab rests on a frictionless floor as shown in the figure. A 10 kg block rests on the top of the slab. The static coefficient of friction between the block and slab is 0.60 while the kinetic friction is 0.40. The 10 kg block is acted upon by a horizontal force 100 N. If g = 9.8 m/s², the resulting acceleration of the slab will be 

(1) 0.98 m/s²
(2) 1.47 m/s²
(3) 1.52 m/s²
(4) 6.1 m/s²

Block A weighing 100 kg rests on a block B and is tied with a horizontal string to the wall at C. Block B weighs 200 kg. The coefficient of friction between A and B is 0.25 and between B and the surface is 1/3. The horizontal force P necessary to move the block B should be (g = 10 m/s²)

(1) 1150 N
(2) 1250 N
(3) 1300 N
(4) 1420 N

A rough vertical board has an acceleration ‘a’ so that a 2 kg block pressing against it does not fall. The coefficient of friction between the block and the board should be

(1) > g/a
(2) < g/a
(3) = g/a
(4) > a/g

A car is negotiating a curved road of radius R. The road is banked at angle $\mathrm{\theta }$. The coefficient of friction between the tyres of the car and the road is ${\mathrm{\mu }}_{\mathrm{s}}$. The maximum safe velocity on this road is
1. $\sqrt{\mathrm{gR}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$
2. $\sqrt{\frac{\mathrm{g}}{\mathrm{R}}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$
3. $\sqrt{\frac{\mathrm{g}}{{\mathrm{R}}^2}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$
4. $\sqrt{{\mathrm{gR}}^2\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$
 

What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?
(1) $\sqrt{2\mathrm{gR}}$
(2) $\sqrt{3gR}$
(3) $\sqrt{5gR}$
(4) $\sqrt{gR}$
 

A particle of mass 10g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration, if the kinetic energy of the particle becomes equal to 8x10^-4 J by the end of the second revolution after the beginning of the motion?
(1) 0.15 m/s²
(2) 0.18 m/s²
(3) 0.2 m/s²
(4) 0.1 m/s²

A block A of mass m₁ rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of table and from its other end another block B of mass m₂ is suspended. The coefficient of kinetic friction between the block and the table is  μ_k. When the block A is sliding on the table, the tension in the string is
1. (m₂+μ_km₁)g /(m₁+m₂)
2. (m₂-μ_km₁)g/(m₁+m₂)
3. m₁m₂(1+μ_k)g/(m₁+m₂)
4. m₁m₂(1-μ_k)g/(m₁+m₂)

A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches 30°, the box starts to slip and slides 4.0 m down the plank in 4.0 s. The coefficients of static and kinetic friction between the box and the plank will be. respectively

1. 0.6 and 0.6
2. 0.6 and 0.5
3. 0.5 and 0.6
4. 0.4 and 0.3

Two stones of masses m and 2m are whirled in horizontal circles, the heavier one in a radius r/2 and the lighter one in radius r. The tangential speed of lighter stone is n times that of the value of heavier stone when they experience same centripetal forces. The value of n is
1. 2
2. 3
3. 4
4. 1

A system consists of three masses m₁, m₂ and m₃ connected by a string passing over a pulley P. The mass $m_1$ hangs freely and m₂ and m₃ are on a rough horizontal table (the coefficient of friction=μ) The pulley is frictionless and of negligible mass. The downward acceleration of mass m₁, is (Assume,m₁=m₂=m₃=m)

1. g(1-gμ)/9
2. 2gμ/3
3. g(1-2μ)/3
4. g(1-2μ)/2

A balloon with mass m is descending down with an acceleration a (where a < g). How much mass should be removed from it so that it starts moving up with an acceleration a?
1. 2ma/g+a
2. 2ma/g-a
3. ma/g+a
4. ma/g-a

The upper half of an inclined plane of inclination θ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and lower half of the plane is given by
1. μ=1/tanθ
2. μ=2/tanθ
3. μ=2tanθ
4. μ=tanθ

A conveyor belt is moving at a constant speed of 2 m/s. A box is gently dropped on it. The coefficient of friction between them is $\mathrm{\mu }=0.5.$ The distance that the box will move relative to the belt before coming to rest on it taking $\mathrm{g}=10 {\mathrm{ms}}^{-2}$, is 
1. $1.2 \mathrm{m}$                                            2. $0.6 \mathrm{m}$
3. zero                                               4. $0.4 \mathrm{m}$

A block of mass m is in contact with the cart C as shown in the figure.

The coefficient of static friction between the block and the cart is $\mu .$The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies
1. $\alpha >\frac{mg}{\mu }$                                       2. $\alpha >\frac{g}{\mu m}$
3. $\alpha \ge \frac{g}{\mu }$                                          4. $\alpha <\frac{g}{\mu }$

A gramophone record is revolving with an angular velocity $\omega .$ A coin is placed at a distance r from the centre of the record. The static coefficient of friction is $\mu .$ The coin will revolve with the record if: 
1. $r=\mu g{\omega }^2$                                       2. $r<\frac{{\omega }^2}{\mu g}$
3. $r\le \frac{\mu g}{{\omega }^2}$                                         4. $r\ge \frac{\mu g}{{\omega }^2}$

Sand is being dropped on a conveyor belt at the rate of M kg/s. The force necessary to keep the belt moving with a constant velocity of v m/s will be 
1. Mv newton 
2. 2Mv newton 
3. $\frac{Mv}{2}$newton 
4. zero