DYNAMICS
LAWS OF MOTION
Force and inertia – Newton’s Laws of Motion. Conservation of linear momentum and its applications, rocket propulsion, friction – laws of friction.
WORK, ENERGY AND POWER
Concept of work, energy and power. Energy –kinetic and potential. Conservation of energy and its applications, Elastic collisions in one and two dimensions. Different forms of energy.
INITIAL STEP EXERCISE
1. In inertial frame of reference (a) (b) (c) (d) 2. (a) (c) 3. Newton’s first law is valid Newton’s second law is valid Newton’s first & second laws are valid Newton’s first, second & third laws are valid Inertial Both (b) (d) Non-inertial cannot be said
The frame of reference attached with the earth is
A particle of mass m moving with speed u along a straight line path is stopped by a constant force F. In this process the particle moves a distance (a) (c)
mu 2 2F 2mu 2 F
(b) (d)
mu 2 F mu 2 4F
4.
A man is standing on a weighing machine placed in a lift. When stationary, his mass is m. Then choose the incorrect statement (a) (b) (c) (d) If the lift moves with constant velocity then machine record the weight mg. If the lift moves upward with constant acceleration a then machine records the weight m(g + a). If the lift moves downwards with constant acceleration a then machine records the weight m(g – a). If the lift moves upwards with constant retardation a then machine records the weight m(g + a).
5.
A uniform rope of length l is pulled by a constant force F applied at one of the end. The tension in the rope at a distance x from the end where the force is applied as
(a)
F
(b)
⎛ x⎞ F ⎜1 + ⎟ ⎝ l⎠ ⎛ F ⎜1 − ⎝ x⎞ ⎟ l⎠
F
(c) 6.
x l
(d)
A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m by applying a force P at one end of the rope. The force exerted by the rope on the block is (a)
PM M−m Pm M+m
(b)
PM M+m Pm M−m
(c) 7.
(d)
Three blocks each of mass 0.6 kg are connected by two strings as shown in figure
The horizontal surface is smooth and a force of 7.2 N is applied. The tension in the string 1 is (a) (c) 8. 1.2 N 3.6 N (b) (d) 2.4 N 7.2 N
A body takes n times as much time to slide down at 450 rough incline as it takes to slide down a smooth 450 incline. The coefficient of friction is
1−
(a)
1 n2
1 n2
(b)
1 1− n2
1 1− n2
1−
(c) 9.
(d)
A block of mass 2 kg rests on a rough inclined plane making an angle of 300 with the horizontal. The coefficient of static friction between the block and the plane is 0.7. The frictional force on the block is (a) (c) 9.8 N 9.8 × √3 N (b) (d) 0.7 × 9.8 N 0.7 × 9.8 × √3 N
10.
A uniform chain of length L lies on a table. If the coefficient of friction is µ, then the maximum length of the chain which can hang from the edge of the table without the chain sliding down is (a) (c)
L µ µL µ +1
(b) (d)
L µ −1 µL µ –1
11.
A block placed on an inclined plane of slope angle θ slides down with a constant speed. The coefficient of kinetic friction is equal to (a) (c) sin θ tan θ (b) (d) cos θ cot θ
12.
A linear momentum p of a body moving in one dimension varies with time according to the equation p = a + b t2, where a and b are positive constants. The net force acting on the body is (a) (b) (c) proportional to t2 proportional to t a constant
(d) 13.
inversely proportional to t
A block of weight 5N is pushed against a wall by a force of 12N. The coefficient of friction between the wall and the block is 0.6. The magnitude of the force exerted by the wall on the block is (a) (c) 5N 13 N (b) (d) 12 N 15.6 N
14.
In the previous problem, the angle made by the net contact force with normal reaction exerted by the wall on the block is (a) (c) 370 75
0
(b) (d)
530 none of these
15.
A block of mass 1 kg lies on a horizontal surface in a truck, the coefficient of friction between the block and the surface is 0.6. The force of friction on the block if the acceleration of the truck is 5 m/s2 is (a) (c) 2N 4N (b) (d) 3N 5N
16.
A bob is hanging from the ceiling of a car using a mass less string. If the car moves with an acceleration a towards right, the angle made by the string with the vertical is
(a)
⎛a⎞ sin −1 ⎜ ⎟ ⎜g⎟ ⎝ ⎠ , deflected towards right
⎛g⎞ sin −1 ⎜ ⎟ ⎝ a ⎠ , deflected towards left
(b)
(c)
⎛a⎞ tan −1 ⎜ ⎟ ⎜g⎟ ⎝ ⎠ , deflected towards left
⎛g⎞ tan −1 ⎜ ⎟ ⎝ a ⎠ , deflected towards right
(d) 17.
A stream of water of density ρ flowing horizontally with a speed of v gushes out of a tube of cross sectional area a and hits a vertical wall. The force exerted on the wall by the impact of water, assuming that it does not rebound, is (a) (c) ρ av2 2 ρav2 (b) (d) √2 ρav2 4 ρav2
18.
In the previous problem, let the power exerted by the stream of water on the wall is directly proportional to speed v as vn. The value of n is (a) (c) 1 3 (b) (d) 2 4
19.
An impulse is supplied to a moving object with the force at an angle of θ with the velocity vector. The angle between the impulse vector and the change in momentum vector is (a) (c) θ θ/2 (b) (d) 0 2θ
20.
A disc of mass 10 g is kept floating horizontally by throwing 10 marbles per second against it from below. The marble strike the disc normally and rebound downward with the same speed. If the mass of each marble is 5 g, the velocity with which the marble are striking the disc is (g = 9.8 m/s2) (a) (c) 0.98 m/s 1.96 m/s (b) (d) 9.8 m/s 19.6 m/s
21.
A cricket ball of mass 150 g is moving with a velocity of 12 m/s and is hit by a bat so that if is turned back with velocity of 20 m/s. The force of blow acts for 0.01 s. The average force exerted by the bat on the ball is
(a) (c) 22.
120 N 480 N
(b) (d)
240 N 960 N
The magnitude of the force (in newtons) acting on a body varies with time t (in micro-seconds) as shown in the figure. AB, BC and CD are straight line segments. The magnitude of the total impulse of the force on the body from t = 4 µs to t = 16 µs is (a) (c) 5 mili Ns 5 Ns (b) (d) 10 mili Ns 10 Ns
23.
A ball of mass m is thrown upward with a velocity v. If air exerts an average resisting force F, the velocity with which the ball returns back to the thrower is
v
(a)
mg mg + F mg − F mg + F
v
(b)
f mg + F
v
(c) 24.
(d)
none of these
An elastic string of unstretched length L and force constant k is stretched by a small length x. It is further stretched by another small length y. The work done in the second stretching is (a) (c)
1 2 ky 2 1 k ( x + y) 2 2
(b) (d)
1 k(x 2 + y2 ) 2 1 ky(2x + y) 2
25.
A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to (a) (c) t1/2 t
3/2
(b) (d)
t3/4 t2
26.
A uniform chain of length L and mass M is lying on a smooth table and 1/n of its length is hanging vertically down over the edge of the table. The work required to pull the hanging part on the table is (a) (c) MgL (b) (d)
MgL n MgL 2n 2
MgL n2
27.
A cord is used to lower vertically a block of mass M a distance d at a constant downward acceleration of g/4. Then the work done by the cord on the block is (a) (c)
Mgd 4 3Mgd 4
(b) (d)
− Mgd 4 − 3Mgd 4
28.
The displacement x of a particle of mass m kg moving in one dimension, under the action of a force, is related to the time t by the equation t = √x + 3 where x is in metres and t is in seconds. The work done by the force in the first six seconds in joules is (a) (c) 0 6m (b) (d) 3m 9m
29.
A simple pendulum of length l has a bob of mass m. It is displaced through an angle of θ from the vertical and then released. Choose the incorrect option (a) (b) (c) (d) The speed of the bob at the lowest most point is
2gl (1 − cosθ)
Tension in the string at the lowest most point is (3 – 2 cosθ)mg Tension in the string at the point where it is released is 0 The centripetal force at the point where it is released is zero.
30.
A pump can take out 36000 kg of water per hour from a 100 m deep well. It has efficiency of 50%, its power is (g = 10 m/s2). (a) (c) 5 kW 15 kW (b) (d) 10 kW 20 kW
31.
A car moves at a constant speed on a road as shown in figure. The normal force by the road on the car is NA, NB, NC and ND when it is at the points A, B, C and D respectively.
(a) (b) (c) (d) 32.
NA = NB = NC = ND NA > NB > NC > ND NA ND > NA > NB
A simple pendulum has a string of length l and bob of mass m. When the bob is at its lowest position, it is given the minimum horizontal speed necessary for it to move in a circular path about the point of suspension. The tension in the string at the lowest position of the bob is (a) (c) 3 mg 5 mg (b) (d) 4 mg 6 mg
33.
The work done in moving a particle from a point (1, 1) to (2, 3) in a plane and in a force field with potential U = λ (x + y) is (a) (c) 0 3λ 2% 6% 20% 44% (b) (d) (b) (d) (b) (d) λ –3λ 4% 8% 40% 50%
34.
If the momentum of the body is increased by 2% then the percentage increase in its kinetic energy is (a) (c)
35.
If the momentum of the body is increased by 20% then the percentage increase in its kinetic energy is (a) (c)
36.
A particle of mass 0.1 kg (moving along x-axis) is subjected to a force, which varies with distance (x) as shown in figure. It starts journey from rest at x = 0, its speed at x = 12 m is
(a) (c) 37.
20 m/s 60 m/s
(b) (d)
40 m/s 80 m/s
a b − 6 12 r r ; The potential energy between two atoms in a diatomic molecule can be expressed as where a and b are constants and r is the distance between the atoms. The conservative force function F is given by U(r ) =
(a) (c)
12a 6b + r13 r 7 12a 6b − r13 r 7
(b) (d)
− 12a 6b + 7 r13 r − 12a 6b − 7 r13 r
38.
A particle is projected with speed u at an angle θ with the horizontal in a vertical plane. The instantaneous power of the particle at the instant when it at the highest point of its trajectory is (a) (c) 0 mgu sinθ (b) (d) mgu cosθ mgu tanθ
39.
In the above problem, the average power during the time from point of projection to the instant when the particle is at the highest point of its trajectory is (a) (c) 0 –½mgu sinθ (b) (d) –½mgu cosθ –½mgu tanθ
40.
A small block of mass 0.1 kg moves with uniform speed in a horizontal circular groove, with vertical side walls, of radius 0.25 m. If the block takes 2 s to complete one round, the normal contact force by the side wall of the groove on the block is
(a)
π2 N 40 π N 10
2
(b)
π2 N 20 π2 N 5
(c) 41.
(d)
The speed at which a car can run round a curve of 30 m radius on a level road if the coefficient of friction between the tyres and the road is 0.4, is (a) (c) 5 m/s 20 m/s (b) 10 m/s (d) none
42.
An ideal spring of force constant k is attached to a vertical wall as shown in figure. A block of mass m is projected with speed u towards the spring. The horizontal surface is smooth. The maximum compression in the spring is
2u m k u m k
(a)
(b)
(c)
u m 2 k
(d)
u m 4 k
43.
In the above problem, the speed of the block when it compress the spring by an amount half of the maximum compression is (a) (c) u/2 u/√2 (b) (d) u/4 none of these
44.
A particle of mass m is moving in a circle of radius r center at O with constant speed v. The magnitude of change in linear momentum in moving from A to B is
2mv sin
(a) (c) 0 (b)
θ 2 θ 2
2mv cos
θ 2
2mv tan
(d)
45.
A ball A, moving with a speed u, collides directly with another similar ball B moving with a speed v in the opposite direction. A comes to rest after the collision. If the coefficient of restitution is e then u/v is (a)
1+ e 1− e
(b)
1− e 1+ e
(c)
e 1− e
(d)
e 1+ e
46.
A ball is dropped from a height of 1 m. If the coefficient of restitution between the surface and the ball is 0.6, the ball rebounds to a height of (a) (c) 0.6 m 0.16 m (b) (d) 0.4 m 0.36 m
47.
A ball, moving with a speed v towards north, collides with an identical ball, moving with a speed v towards east. After collision the two balls stick together and move towards north-east. The speed of the combination is (a) (c) v v/√2 (b) (d) v√2 v/2
48.
A radioactive nucleus of mass number A, initially at rest, emits an α particle with speed v. The recoil speed of the daughter nucleus is (a) (c)
4v A−4 ( A − 4) v A
(b) (d)
4v A ( A − 4) v 4
49.
A neutron collides head-on and elastically with an atom of mass number A, which is initially at rest. The fraction of kinetic energy retained by the neutron is
(a)
⎛ A ⎞ ⎟ ⎜ ⎝ A +1⎠
2
(b)
⎛ A −1 ⎞ ⎟ ⎜ ⎝ A +1⎠
2
(c) 50.
⎛ A −1⎞ ⎜ ⎟ ⎝ A ⎠
2
(d)
A −1 A +1
A ball of mass m1 collides head-on and elastically with an identical ball of mass m2, initially at rest. The transfer of energy will be maximum when (a) (b) (c) (d) m1 = m2 m1 = m2/2 m1 = 2m2 none of the above
51.
A ball of mass m collides head-on and elastically with a ball of mass nm, initially at rest. The fraction of the incident energy transferred to the heavier ball is (a) (b) (c) (d)
n n +1
n (n + 1) 2 2n (n + 1) 2
4n (n + 1) 2
52.
An ideal spring with spring constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstreched. Then the maximum extension in the spring is (a) (c)
4Mg k Mg k
jet rocket
(b) (d) (b) (d)
2Mg k Mg 2k
aeroplane all of these
53.
Which of the following works on the conservation of linear momentum? (a) (c)
54.
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 m/s2) (a) (c) 127.5 kg s–1 187.5 kg s–1 (b) (d) 137.5 kg s–1 185.5 kg s–1
55.
A body of mass m1, moving with a velocity u1, collides head-on with a body of mass m2, moving with a velocity u2. The two bodies stick together after the collision. The loss of kinetic energy during the collision is
(a)
m1m 2 (u1 − u 2 ) 2 2(m1 + m 2 ) m 2 m1 (u 2 + u1 ) 2 2(m1 + m 2 )
(b)
(c)
m 2 m1 (u 2 + u1 ) 2 2(m1 − m 2 ) m 2 m1 (u 2 − u1 ) 2 2(m1 − m 2 )
(d) 56.
Keeping the banking angle same, to increase the maximum speed with which a vehicle can travel on a circular road by 10%, the radius of curvature of the road has to be changed from 20 m to (a) (c) 16 m 24.2 m (b) (d) 18 m 30.5 m
Solutions:
1. 7. 13. 19. 25. 31. 37. 43. 49. 55.
a b c b c d c d b a
2. 8. 14. 20. 26. 32. 38. 44. 50. 56.
b a d a d d a b a c
3. 9. 15. 21. 27. 33. 39. 45. 51.
a a d c d c c a d
4. 10. 16. 22. 28. 34. 40. 46. 52.
d c c a a b a d b
5. 11. 17. 23. 29. 35. 41. 47. 53.
d c a c c c d c c
6. 12. 18. 24. 30. 36. 42. 48. 54.
b b c d d b b a c
FINAL STEP EXERCISE
1. Two blocks A and B, having masses m1 and m2 respectively, are placed in contact on a smooth horizontal surface. A force F is applied horizontally on A as shown in figure (I). Let the contact force between A and B is F1. Now the same force is applied horizontally on B as shown in figure (II). Let the contact force between them is F2 in this case. Then F1/F2 equals to
(a)
1:1
(b)
m2 m1 m2 m1 + m 2
(c) 2.
m1 m2
(d)
There are N identical blocks each of mass m are placed as shown in figure. Assume that the surface is smooth and a force F is applied on the block 1.
Choose the correct statement from the following (a) The acceleration of any block is F/mN
(b)
2⎞ ⎛ ⎜1 − ⎟ F N⎠ The interaction force between second and third block is ⎝
3.
4.
F (c) The force exerted by (N – 1)th block on Nth block is N (d) All are correct A monkey of mass 40 kg climbs on a mass less rope of breaking strength 600 N. The rope will break if the monkey (g = 10 m/s2) (a) climbs up with a uniform speed of 5 m/s (b) climbs up with an acceleration of 6 m/s2 (c) climbs down with an acceleration of 4 m/s2 (d) climbs down with a uniform speed of 5 m/s. Three blocks A, B and C, each of mass m, are hanging over a fixed pulley as shown in figure. The tension in the string connecting B and C is (a) zero (b) mg/3 (c) 2mg/3 (d) mg
5.
In the previous problem the force exerted by the string connected to the ceiling on the pulley is (a) zero (b) mg/3
(c) 6. (a) (c)
2mg/3 200 N 100 N
(d) (b) (d)
8mg/3 100√3 N 200√3 N
The tension in the fifth string is
7.
A homogenous rod with a length L is acted upon by two collinear forces F1 and F2 applied to its ends and directed opposite. The magnitude of the pulling force F at the cross-section distant l from F1 end (F1 > F2) is (a) (b) (c) (d)
F1 + F1 + F2 + F2 +
l (F2 − F1 ) L l (F1 − F2 ) L l (F1 − F2 ) L l (F2 − F1 ) L
8.
Two blocks in figure are connected by uniform rope of mass 4 kg. An upward force of 200 N is applied. The net force on the rope is (a) (b) (c) (d) 6.5 N 13.3 N 23.2 N 40.0 N
9.
In the previous problem, tension at the mid point of the rope is (a) (c) 47.8 N 93.3 N (b) (d) 67.2 N 100 N
10.
In the figure shown, all the surface are smooth. The bigger block has given an acceleration a towards right such that m1 and m2 are stationary with respect to bigger block. The value of a is
(a)
m1g m2
(b)
m 2g m1
(c)
m1g m1 + m 2
(d)
g
11.
In the previous problem the force exerted by the bigger block on m1 is
2 m1 g m1 + m 2 2 m1 g m2
(a)
(b)
(c) 12.
mg m1
2 2
(d)
m 2g 2 m1 + m 2
A smaller block of mass m is placed on the wedge of mass M. All the surfaces are smooth. The magnitude of the horizontal force F required to keep the block at rest with respect to wedge is (a) (c) (M + m)g sin θ (b) (M + m)g cot θ (d) (M + m)g cos θ (M + m)g tan θ
13.
Figure shows two blocks m and M, which are pushed together on a smooth horizontal surface. If the coefficient of friction between the blocks is µ then the minimum value of the horizontal force to hold the blocks together is (a) (c)
( M + m )g µ ( M + m )g 3µ
(b) (d)
( M + m )g 2µ ( M + m )g 4µ
14.
A block of mass m is placed on another block of mass M lying on a smooth horizontal surface. The coefficient of static friction between m and M is µ. The maximum force that can be applied to m so that the blocks remain at rest relative to each other is (a) (c)
µ(m + M)mg 2M µ(m + M)mg M
(b) (d)
µ(m + M)mg 3M µ(m + M)mg 4M
15.
A block A of mass 200 kg rests on a block B of mass 300 kg. A is tied with a horizontal string to a wall. Coefficient of friction between A and B is 0.25 and heat between B and floor is 0.2. The horizontal force F needed to move the block B is (g = 10 m/s2) (a) (c) 550 N 1500 N (b) (d) 1100 N 2200 N
16.
The total mass of an elevator, with an 80 kg man in it, is 1000 kg. It is moving upwards with a speed of 8 m/s. If it is brought to rest over a distance of 16 m, then during retardation, the tension in the supporting cable and the force exerted by the man on the floor are respectively (g = 10 m/s2) (a) (c) 4000 N, 640 N 4000 N, 320 N (b) (d) 8000 N, 640 N 8000 N, 320 N
17.
A painter is raising himself and the crate on which he stands with an acceleration of 5 m/s2 by a mass less rope-and-pulley arrangement. Mass of the painter is 100 kg and that of the crate is 50 kg. If g = 10 m/s2, then the (a) (b) (c) (d) tension in the rope is 2250 N tension in the rope is 1725 N force of contact between the painter and the floor is 750 N force of contact between the painter and the floor is 375 N.
18.
In a simple pendulum, the breaking strength of the string is double the weight of the bob. The bob is released from rest when the string is horizontal. The string breaks when it makes an angle θ with the vertical. (a) (c)
θ = cos–1(1/3)
(b)
θ = 600
19.
θ = cos–1(2/3) (d) θ=0 A particle of mass m is fixed to one end of a light rigid rod of length 1 and rotated in a vertical circular path about its other end. The minimum speed of the particle at its lowest point must be
(a) (b) zero (b) (d)
gl 2 gl
1.5 gl
20.
A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration a is varying with time as ac = k2rt2, where k is a constant. The power delivered to the particle by the forces acting on it is (a) (c) 2πmk2r2t mk4r2t5 (b) (d) mk2r2t 0
21.
Under the action of a force, a 2 kg body moves such that the position x as a function of time t is given by x = 2t3, where x is in metres and t is in seconds. The work done by the force in the first three seconds is (a) (c) 144 J 1660 J (b) (d) 720 J none of these
22.
A block of mass m is pulled along a horizontal surface by applying a force at an angle θ with the horizontal. The friction coefficient between the block and the surface is µ. If the block travels at a uniform speed, the work done by this applied force during a displacement d of the block is (a) (c)
µMgd cos θ cos θ + µ sin θ
(b)
µMgd sin θ cos θ + µ sin θ
23.
(d) r ˆ + xˆ) A force F = − k ( yi j , where k is a positive constant, acts on a particle moving in the xy plane. Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0), and then parallel to the y-axis to the point (a, a). The total work done by the force on the particle is (a) (c) 0 –ka2 (b) (d)
1/ 3
µMgd sin θ sin θ + µ cos θ
µMgd cos θ sin θ + µ cos θ
ka2 –2ka2
24.
The speed v reached by a car of mass m that is driven with constant power P is given by
(a)
⎛ 3xP ⎞ ⎜ ⎟ ⎝ m ⎠
(b)
1/ 2
⎛ 2xP ⎞ ⎜ ⎟ ⎝ m ⎠ ⎛ 3xP ⎞ ⎜ ⎟ ⎝ m ⎠
1/ 3
(c) 25.
⎛ 2xP ⎞ ⎜ ⎟ ⎝ m ⎠
1/ 2
(d)
where x is the distance traveled by car from rest. A small particle of mass m attached to a string of length l is rotated about a vertical axis at constant angular speed ω, as shown in the figure. The value of ω is
g l sin θ
g lcotθ
(a)
(b)
g ltanθ
g lcosθ
(c)
(d)
26.
A small block of mass m moving on the inside of a smooth fixed hollow hemisphere of radius r, describes a horizontal circle at a distance of r/2 below the centre of the sphere. The force with which the block pushes against the hemisphere is (a) (c) mg 3mg (b) (d) 2mg
√3mg/2
27.
A canon ball is fired with a velocity of 200 m/s at an angle of 600 with the horizontal. At the highest point it explodes into three equal fragments. One goes vertically upwards with a velocity of 100 m/s, the second one falls vertically downwards with a velocity of 100 m/s. The third one moves with a velocity of (a) (b) (c) (d) 100 m/s horizontally 300 m/s horizontally 200 m/s at 600 with the horizontal 300 m/s at 600 with the horizontal
28.
A body of mass 2.9 kg is suspended from a string of length 2.5 m and is at rest. A bullet of mass 100 g, moving horizontally with a speed of 150 m/s, strikes and sticks to it. The maximum angle made by the string with the vertical after the impact is (a) (c) 300 600 (b) (d) 450 900
29.
A 1 kg ball, moving at 12 m/s, collides head-on with a 2 kg ball moving in the opposite direction at 24 m/s. If the coefficient of restitution is 2/3, then the energy lost in the collision is (a) (c) 60 J 240 J (b) (d) 120 J 480 J
30.
A loaded spring gun, initially at rest on a horizontal frictionless surface, fires a marble at angle of elevation θ. The mass of the gun is M, the velocity of the marble is v0 with respect to gun. The velocity of the gun after the fire is (a) (c)
mv0 cos θ m+M Mv 0 cos θ m
⎛m+M⎞ ⎜ ⎟ tan θ ⎝ m ⎠ ⎛m+M⎞ ⎜ ⎟ tan θ ⎝ M ⎠
(b) (d)
mv0 cos θ M mv0 sin θ M
31.
In the above problem, let the angle of elevation as seen from the ground is α then tan α equals to
(a)
(b) (c)
tan θ
⎛m+M⎞ ⎜ ⎟ cos θ ⎝ M ⎠
(d) 32.
A 20.0 kg body is moving in the positive x-direction with a speed of 200 m/s when, owing to an internal explosion, if breaks into three parts. One part, with a mass of 10.0 kg moves away from the point of explosion with a speed of 100 m/s in positive y direction. A second fragment, with a mass of 4.0 kg moves in the negative x-direction with a speed of 500 m/s. The energy released in the explosion is (a) (c) 1.25 MJ 3.23 MJ (b) (d) 2.5 MJ 4.75 MJ
33.
A ball collides with a fixed plane with initial speed u at an angle α with the normal. The coefficient of restitution for the collision is e. The sphere rebounds with speed v at an angle β with normal. Then tan β equals to (a)
sin α e
(b)
cosα e
(c) 34.
cotα e
(d)
tan α e
Two equal balls of mass m are in contact on a smooth horizontal table. A third identical ball with velocity u collides symmetrically on them and comes to rest. The coefficient of restitution for the collision is (a) (c) 1/3 2/5 (b) (d) 2/3 1/5
35.
Two equal spheres A and B lie on a smooth horizontal circular groove at opposite ends of diameter. A is projected along the groove and at the end of time t impinges on B. If e is the coefficient of restitution then the time after which the second impact will occur (a) (c) t/e 3t/e (b) (d) 2t/e 4t/e
36.
A ball is dropped from a height h onto a floor. If in each collision its speed becomes e times of its striking value. The time taken by the ball to stop rebounding
(a)
2h ⎛ 1 − e ⎞ ⎜ ⎟ g ⎝1+ e ⎠ h ⎛1+ e ⎞ ⎜ ⎟ g ⎝1− e ⎠
(b)
h ⎛1− e ⎞ ⎜ ⎟ g ⎝1+ e ⎠ h ⎛1+ e ⎞ ⎜ ⎟ g ⎝1− e ⎠
(c) 37.
(d)
A perfectly elastic oblique collision takes place between a moving particle and a stationary particle of the same mass. After the collision the angle between their directions of motion is (a) (c)
π/3
(b)
π/4
38.
π/2 (d) cannot be determined A block of mass m is projected with speed u as shown in figure towards the block of mass 2m. A spring of force constant k is connected to the block of mass 2m. Assume all the surfaces are frictionless. At the maximum compression of the spring the velocity of each block are given by
(a) (c) 39.
u/2, u/2 u/2, u/3
m k m k
(b) (d)
u/3, u/3 u/3, u/2
m k
In the above problem the maximum compression of the spring is
2u u
(a)
u 2
(b)
(c) 40.
(d)
none of these
Four particles P, Q, R and S of equal mass move with equal speed v along the diagonals of square as shown in figure. They collide at the center O of the square. After the collision, P comes to rest and S retraces its path with speed v. Then it is possible that (a) (b) (c) (d) R comes to rest whereas Q retraces its path with speed v. Both Q and R comes to rest. Q comes to rest whereas R retrices its path with speed v. Both Q and R retraces its path with speed v.
41.
A ball is held at a rest in position A by two light cords. The horizontal cord is now and the ball swings to the position B. What is the ratio of the tension in the cord in position B to that in position A. (a) (b) (c) (d) 3 3/4 1/2 1
Solutions:
1. 7. 13. 19. 25. 31. 37.
b a a d d b c
2. 8. 14. 20. 26. 32. 38.
d b c b c c b
3. 9. 15. 21. 27. 33. 39.
b c c d b d d
4. 10. 16. 22. 28. 34. 40.
c a b a c b a
5. 11. 17. 23. 29. 35. 41.
d b d c c b b
6. 12. 18. 24. 30. 36.
a d c a a d
ANALYSIS
1. A light spring balance hangs from the hook of the other light spring balance and a block of mass M kg hang from the former one. Then the true statement about the scale reading is (a) (b) (c) (d) [Ans. : c] 2. A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m. If a force P is applied at the free end of the rope, the force exerted by the rope on the block is (a) (c) [Ans. : b] 3. A horizontal force of 10 N is necessary to just hold a block stationary against a wall. The coefficient of friction between the block and the wall is 0.2. The weight of the block is P (b) (d) The reading of the two scales can be anything but the sum of the reading will be M kg Both the scales read M/2 kg each Both the scales read M kg each The scale of the lower one reads M kg and of the upper one zero
PM M+m Pm M−m
Pm M+m
(a) (c) [Ans. : b] 4.
100 N 20 N
(b) (d)
2N 50 N
A marble block of mass 2 kg lying on ice when given a velocity of 6 m/s is stopped by friction in 10 s. Then the coefficient of friction is (a) (c) [Ans. : ] 0.04 0.02 (b) (d) 0.01 0.03
5.
Consider the following two statements A. B. Then (a) (b) (c) (d) [Ans. : a] A does not imply B but B implies A A implies B and B implies A A does not imply B and B does not imply A A implies B but B does not imply A Linear momentum of a system of particles is zero Kinetic energy of a system of particles is zero
6.
A body is moving along a straight line by a machine delivering a constant power. The distance moved by body in time ‘t’ is proportional to
(a) (c) [Ans. : d] 7.
t1/4 t3/4
(b) (d)
t1/2 t3/2
A rocket with a lift-off mass 3.5 × 104 kg is blasted upwards with an initial acceleration of 10 m/s2. Then the initial trust of the blast is (a) (c) [Ans. : c] 14.0 × 105 N 3.5 × 105 N (b) (d) 1.75 × 105 N 7.0 × 105 N
8.
A car, moving with a speed 50 km/hr can be stopped by brakes after at least 6 m. If the same car is moving at a speed of 100 km/hr, the minimum stopping distance is (a) (c) [Ans. : a] 24 m 12 m (b) (d) 6m 18 m
9.
A spring balance is attached to the ceiling of a lift. A man hangs his bag on the spring and the spring reads 49 N, when the lift is stationary. If the lift moves downward with an acceleration of 5m/s2, the reading of the spring balance will be (a) (c) [Ans. : c] 15 N 24 N (b) (d) 49 N 74 N
10.
When a U238 nucleus originally at rest, decays by emitting an alpha particle having a speed ‘u’ the recoil speed of the residual nucleus is (a) (c)
4u 234 4u 238
−
(b)
4u 238 4u 234
−
(d)
[Ans. : d] 1. A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to (a) (c) [Ans. : a] 2. An automobile travelling with a speed of 60 km/h, can brake to stop within a distance of 20 m. If the car is going twice as fast, i.e. 120 km/h, the stopping distance will be (a) (c) [Ans. : d] 3. A machine gun fires a bullet of mass 40 g with a velocity 1200 ms–1. The man holding it can exerts a maximum force of 144 N on the gun. How many bullets can be fire per second at the most? (a) (c) [Ans. : d] 4. Two mass m1 = 5 kg and m2 = 4.8 kg tied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when lift free to move? (g = 9.8 m/s2) (a) 0.2 m/s2 one two (b) (d) four three 20 m 60 m (b) (d) 40 m 80 m x2 x (b) (d) ex logex
(b) (c) (d)
9.8 m/s2 5 m/s2 4.8 m/s2
[Ans. : a] 5. A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table? (a) (c) [Ans. : b] 6. A block rests on a rough inclined plane making an angle of 300 with the horizontal. The coefficient of static friction between the block and the plane is 0.8. If the frictionless force on the block is 10 N, the mass of the block (in kg) (take g = 10 m/s2) (a) (c) [Ans. : a] 7. 2.0 1.6 (b) (d) 4.0 2.5 7.2 J 120 J (b) (d) 3.6 J 1200 J
ˆ ˆ ˆ A force F = 5i + 3 j + 2k N is applied over a particle which displaces it from its origin to the point r r = ( 2ˆ − ˆ) m the work done on the particle in joules is i j
(a) (c) –7 +10 (b) (d) +7 +13
[Ans. : b] 8. A body of mass m, accelerates uniformly form rest to v1 in time t1. The instantaneous power delivered to the body as a function of time t is
(a)
mv 1t t1
(b)
2
mv1 t 2 t1
2
(c) [Ans. : b] 9.
mv1t t1
(d)
mv1 t t1
2
A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle; the motion of the particle takes place in a plane. It follows that (a) (b) (c) (d) its velocity is constant its acceleration is constant its kinetic energy is constant it moves in a straight line
[Ans. : c] 10. A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion? (a) (c) [Ans. : b] 11. The block of mass M moving on the frictionless horizontal surface collides with the spring of spring constant K and compresses it by length L. The maximum momentum of the block after collision is (a) (b) (c) (d) zero 1.5 cm 3.0 cm (b) (d) 1.0 cm 2.0 cm
ML2 K
MK L
KL2 2M
[Ans. : c] 12. A body of mass m is accelerated uniformly from rest to a speed v in a time T. The instantaneous power delivered to the body as a function of time is given by (a) (c) [Ans. : c] 13. Consider a car moving on a straight road with a sped of 100 m/s. The distance at which car can be stopped is [µk = 0.5] (a) (c) [Ans. : d] 14. A smooth block is released at rest on a 450 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 100 m 800 m (b) (d) 400 m 1000 m
1 mv 2 .t 2 T2
mv .t T2
2
(b) (d)
1 mv 2 2 .t 2 T2
mv 2 2 .t T2
(a)
µs = 1 − µk = 1 −
1 n2 1 n2
µs = 1 −
(b)
µk = 1 −
1 n2
(c) [Ans. : c] 15.
(d)
1 n2
The upper half of an inclined place with inclination φ 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 (a) (c) 2 tan φ 2 sin φ (b) (d) tan φ 2 cos φ
[Ans. : a] 16. A particle of mass 0.3 kg is subjected to a force F = –kx with k = 15 N/m. What will be its initial acceleration if it is released from a point 20 cm away from the origin? (a) (c) [Ans. : b] 17. A block is kept on a frictionless inclined surface with angle of inclination ‘α’. The incline is given an acceleration ‘a’ to keep the block stationary. Then a is equal to 5 m/s2 3 m/s
2
(b) (d)
10 m/s2 15 m/s2
(a) (c) [Ans. : b]
g g/ tan α
(b) (d)
g tan α g cosec α