PARTICLE DYNAMICS

NEWTONS LAWS, FORCE & FRICTION

KEY CONCEPT Page –2
EXERCISE–I Page –4
EXERCISE–II Page –8
EXERCISE–III Page –10
ANSWER KEY Page –16

CIRCULAR MOTION & WORK POWER ENERGY

KEY CONCEPT Page –17
EXERCISE–I Page –19
EXERCISE–II Page –22
EXERCISE–III Page –23
ANSWER KEY Page –29

CENTRE OF MASS, MOMENTUM & COLLISION

KEY CONCEPT Page –30
EXERCISE–I Page –32
EXERCISE–II Page –36
EXERCISE–III Page –38
ANSWER KEY Page –43

VIBRANT ACADEMY (India) Private Limited

Believe In Excellence
B-41, Road No.2, Indraprastha Industrial Area, Kota-324005 (Raj.)
Tel. : 06377791915, (0744) 2778899, Fax : (0744) 2423405
Email: admin@vibrantacademy.com Website : www.vibrantacademy.com
Website : dlp.vibrantacademy.com
 NEWTONS LAW FORCE & FRICTION
KEY CONCEPT
FORCE
1. There are, basically, five forces, which are commonly encountered in mechanics.
(i) Weight : Weight of an object is the force with which earth attracts it. It is also called the force of gravity or
the gravitational force.

(ii) Contact Force : When two bodies come in contact they exert forces on each other that is called contact
forces.
(a) Normal force (N) : It is the component of contact force normal to the surface. It
measures how strongly the surfaces in contact are pressed together.
(b) Frictional force : It is the component of contact force parallel to the surface.
It opposes the relative motion (or attempted motion) of the two surfaces in contact.

(iii) Tension : The force exerted by the end of a taut string, rope or chain is called the tension. The direction of
tension is to pull the body while that of normal reaction is to push the body.

(iv) Spring force : The force exerted by a spring is given by F = – kx, where x is the change in length and k is
the stiffness constant or spring constant (units Nm–1).

NEWTON'S LAWS
2. Newton's First Law : Every particle continues in its state of rest or of uniform motion in a straight line unless
it is compelled to change that state by the action of an applied force.
r r
3. Newton's Second Law : Fnet = m a

4. Newton's Third Law : Whenever two bodies interact they exert forces on each other which are equal in magnitude
and opposite in direction. So whenever body A exerts a force F on body B, B exerts a force – F on A.

Inertial Reference Frame : A reference frame in which Newton’s first law is valid is called an inertial
reference frame. An inertial frame is either at rest or moving with uniform velocity.

Non-Inertial Frame : An accelerated frame of reference is called a non-inertial frame. Objects in non-
inertial frames do not obey Newton’s first law.

Pseudo Force : It is an imaginary force which is recognized only by a non-inertial observer to explain the
physical situation according to Newton’s law. The magnitude of this force FP is equal to the product of the
mass m of the object and acceleration a of the frame of reference. The direction of the force is opposite to the
direction of acceleration.
FP = – ma
The force of friction comes into action only when there is a relative motion between the two contact
surfaces or when an attempt is made to have it.

The force of friction on each body is in a direction opposite to its motion (existing or impending) relative to
other body.
5. Static friction : The frictional force acting between any two surfaces at rest with respect to each other is
called the force of static friction (fs).
fs £ msN
where ms is the static coefficient of friction.
Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [2]
6. Kinetic friction : The frictional force acting between surfaces in relative motion
with respect to each other is called the force of kinetic friction or sliding friction
(fk).
fk = mkN
where mk is the coefficient of kinetic friction.
ms > mk

Angle of friction (f) : Mathematically, the angle of friction (f) may be defined as the angle between the
normal reaction N and the resultant of the maximum friction force f and the normal reaction.

f
Thus tanf =
N
Since f = mN, therefore,
tan f = m

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [3]
 EXERCISE-I
1. The pulley (figure) is fixed, light and smooth. The string is light and mass m < mass M. At t = 0, M and
m are at rest, when the system is released. At time t = t1, the mass ‘M’ is displaced by ‘d’. The speed
of the big block M at time t1 with respect to the pulley is

2(M + m)gd
(A)
M

2 Mgd
(B)
(M + m)

2(M - m)dg
(C)
M+m

2(M - m)dg
(D)
M

2. Three blocks 1, 2 & 3 rest on a horizontal frictionless surface, as shown. Each block has a mass m and the
blocks are connected by massless strings. Block 3 is pulled to the right by a force F. The resultant force on
block 2 is:
(A) Zero (B) (1/3)F
(C) (1/2)F (D) (2/3)F

3. Fig shows two pulley arrangements for lifting a mass m. In (a) the mass is lifted by attaching a mass 2m
while in (b) the mass is lifted by pulling the other end with a downward force F = 2 mg, If fa and fb are the
accelerations of the two masses then
(A) fa = fb
(B) fa = fb/2
(C) fa = fb/3
(D) fa = 2fb

4. A helicopter of mass M is carrying a box of mass m at the end of a rope and is moving
horizontally with constant acceleration 'a'. The acceleration due to gravity is 'g'. Neglect air
resistance. The rope is stretched out from the helicopter at a constant angle q to the
vertical. What is this angle?
(A) sinq = a/g (B) cosq = a/g (C) tanq = a/g (D) sinq = ma/(Mg)

5. Both the blocks shown here are of mass m and are moving with constant
velocity in direction shown in a resistive medium which exerts equal constant
force on both blocks in direction opposite to the velocity. The tension in the
string connecting both of them will be : (Neglect friction)
(A) mg (B) mg/2
(C) mg/3 (D) mg/4

6. Block of 1 kg is initially in equilibrium and is hanging by two identical springs A and

B as shown in figures. If spring A is cut from lower point at t=0 then, find acceleration
of block in ms–2 at t = 0.
(A) 5 (B) 10 (C) 15 (D) 0
7. A stunt man jumps his car over a crater as shown (neglect air resistance)
(A) during the whole flight the driver experiences weightlessness
(B) during the whole flight the driver never experiences weightlessness
(C) during the whole flight the driver experiences weightlessness only at the highest point
(D) the apparent weight increases during upward journey

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [4]
8. A weight can be hung in any of the following four ways by string of same type. In which case is the string
most likely to break?
(A) A (B) B (C) C (D) D

9. A block of mass 2 kg is given a push horizontally and then the block starts sliding over a horizontal plane.
The graph shows the velocity - time graph of the motion. The coefficient of sliding friction between the
plane and the block is

(A) 0.02 (B) 0.20 (C) 0.04 (D) 0.4.

10. A block of mass m is placed on an inclined plane with angle of inclination q. Let N, fL and F respectively
represent the normal reaction, limiting force of friction and the net force down the inclined plane, Let m be the
coefficient of friction. The dependence of N,fL and F on q is indicated by plotting graphs as shown below. Then
curves (1), (2) and (3) respectively represent (µ £ 1)

(A) N, F and fL (B) fL, F and N (C) F, N and fL (D) fL, N and F

11. A horizontal force of 10.0 N is acting on a 10 kg box that is sliding to the right along the floor with velocity v
(as depicted in the adjacent figure). The coefficient of kinetic friction between the box and the floor is 0.20.
The box is moving with
(A) acceleration to the left.
(B) acceleration to the right.
(C) constant speed and constant velocity.
(D) constant speed but not constant velocity.

12. A minimum 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
(A) 100 N
(B) 50 N
(C) 20 N
(D) 2 N

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [5]
13. Coefficient of friction between 5 kg and 10 kg block is 0.5. If friction between them is
20 N. What is the value of force being applied on 5 kg. The floor is
frictionless
(A) 10 N (B) 30 N (C) 20 N (D) 40 N

14. What is the force of friction acting on the 1 kg block placed on the incline as shown in the figure.

(A) 8N (B) 6N (C) 4.8N (D) 6.4N

r
15. A force F = î + 4ˆj acts on block shown. The force of friction acting on the block is :
(A) – î (B) – 1.8 î
(C) – 2.4 î (D) – 3 î

16. A block placed on a rough inclined plane of inclination (q = 30°) can just be pushed
upwards by applying a force "F" as shown. If the angle of inclination of the inclined
plane is increased to (q = 60°), the same block can just be prevented from sliding
down by application of a force of same magnitude. The coefficient
of friction between the block and the inclined plane is

3 +1 2 3 -1 3 -1
(A) (B) (C) (D) None of these
3 -1 3 +1 3 +1

Question No. 17 to 19 (3 questions)

A particle of mass m is constrained to move on x-axis. A force F acts on the
particle. F always points toward the position labeled E. For example, when the
particle is to the left of E, F points to the right. The magnitude of F is a
constant F except at point E where it is zero.
The system is horizontal. F is the net force acting on the particle. The particle is displaced a distance A
towards left from the equilibrium position E and released from rest at t = 0.
17. What is the period of the motion?

æ 2Am ö æ 2Am ö æ 2 Am ö
ç
(A) 4ç
÷ ç
(B) 2ç
÷ (C) ç
ç ÷ (D) None
F ÷ F ÷ F ÷
è ø è ø è ø
18. Velocity – time graph of the particle is

(A) (B)

(C) (D)

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [6]
 A
19. Find minimum time it will take to reach from x = – to 0.
2

3 mA mA
(A) ( 2 - 1) (B) ( 2 - 1)
2 F F

mA
(C) 2 ( 2 - 1) (D) None
F

Match The Column :

1
20. The inclined surface is rough with µ = . For different values of m and
2
M, the system slides down or up the plane or remains stationary.
Match the appropriate entries of column-I with those of column-II.

Column-I Column-II

m 5
(A) Minimum value of so that m slides down (P)
M 3

M
(B) Minimum value of so that m slides up. (Q) 1
m

m 3
(C) Value of so that friction force on m is zero (R)
M 5
(D) Ratio of vertical component of acceleration
of m and acceleration of M. (S) 5

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [7]
 EXERCISE-II
1. Figure shows three blocks in contact and kept on a smooth horizontal surface. What is ratio of force exerted
by block A on B to that of B on C.

2. In the fig. at the free end of the light string, a force F is appled to keep the suspended mass
of 18 kg at rest. Assuming pulley is light then the force exerted by the ceiling on the
system.

3. Block A of mass m/2 is connected to one end of light rope which passes over a pulley
as shown in the Fig. Man of mass m climbs the other end of rope with a relative
acceleration of g/6 with respect to rope find acceleration of block A and
tension in the rope.

4. At what value of m 1 will 8 kg mass be at rest.

5. Block M slides down on frictionless incline as shown. Find the minimum

friction coefficient so that m does not slide with respect to M.

6. Find the acceleration of the blocks and magnitude & direction of frictional
force between block A and table, if block A is pulled towards
left with a force of 50N.

7. Coefficient of friction between 5 kg and 10 kg block is 0.5. If friction between them is

20 N. What is the value of force being applied on 5 kg. The floor is frictionless
8. The diagram shows particles A and B, of masses 0.2 kg and m kg
respectively, connected by a light inextensible string which passes over a
fixed smooth peg. The system is released from rest, with B at a height of
0.25 m above the floor. B descends, hitting the floor 0.5 s later. All
resistances to motion may be ignored.
(a) Find the acceleration of B as it descends.
(b) Find the tension in the string while B is descending and find also the value of m.
(c) When B hits the floor it comes to rest immediately, and the string becomes slack. Find the length of time for
which B remains at rest on the ground before being jerked into motion again.

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [8]
9. A 1kg block ‘B’ rests as shown on a bracket ‘A’ of same mass. Constant
forces F1 = 20N and F2 = 8N start to act at time t = 0 when the distance of
block B from pulley is 50cm. Time when block B reaches
the pulley is _________.

10. A car begins to move at time t = 0 and then accelerates along a straight track with a speed given byV(t) =
2t2 ms–1 for 0 < t < 2
After the end of acceleration, the car continues to move at a constant speed. A small block initially at rest on
the floor of the car begins to slip at t = 1sec. and stops slipping at t = 3 sec. Find the coefficient of static and
kinetic friction between the block and the floor.

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [9]
 EXERCISE-III
1. When forces F1, F2, F3 are acting on a particle of mass m such that F2 and F3 are mutually perpendicular,
then the particle remains stationary. If the force F1 is now removed then the acceleration of the particle is

F1 F2F3 F2 – F3 F2
(A) (B) mF (C) (D) [AIEEE-2002]
m 1 m m

2. Two forces are such that the sum of their magnituedes is 18 N and their resultant is 12 N which is perpendicular
to the smaller force. Then the magnitudes of the forces are [AIEEE-2002]
(A) 12 N, 6 N (B) 13 N, 5 N (C) 10 N, 8 N (D) 16 N, 2 N

3. Speeds of two identical cars are u and 4u at the specific instant. The ratio of the respective distances in
which the two cars are stopped from that instant is [AIEEE-2002]
(A) 1 : 1 (B) 1 : 4 (C) 1 : 8 (D) 1 : 16

4. A light string passing over a smooth light pulley connects two blocks of masses m 1 and m2 (vertically). If the
acceleration of the system is g/8, then the ratio of the masses is [AIEEE-2002]
(A) 8 : 1 (B) 9 : 7 (C) 4 : 3 (D) 5 : 3

5. Three identical blocks of masses m = 2 kg are drawn by a force F = 10.2 N with an acceleration of
0.6 ms–2 on a rough surface, then what is the tension (in N) in the string between the blocks B and C?

(A) 9.2 (B) 7.8 (C) 3.4 (D) 9.8 [AIEEE-2002]

6. One end of a massless rope, which passes over a massless and frictionless pulley P is tied to a hook C
while the other end is free. Maximum tension that the rope can bear is 360 N. With what value of minimum
safe acceleration (in ms–2) can a man of 60 kg climb down the rope? [AIEEE-2002]

(A) 16 (B) 6 (C) 4 (D) 8

7. Two masses m 1 = 5 kg and m 2 = 4.8 kg tied to a string are hanging over a light frictionless pulley. What is the
acceleration of the masses when left free to move? (g = 9.8 m/s2) [AIEEE-2004]
(A) 0.2 m/s2

(B) 9.8 m/s2

(C) 5 m/s2

(D) 4.8 m/s2

8. A block rests on a rough inclined plane making an angle of 30º with the horizontal. The coefficient of static
friction between the block and the plane is 0.8. If the frictional force on the block is 10 N, the mass of the
block (in kg) is (take g = 10 m/s2) [AIEEE-2004]
(A) 2.0 (B) 4.0 (C) 1.6 (D) 2.5

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [10]
9. A player caught a cricket ball of mass 150 g moving at a rate of 20 m/s. If the catching process is completed
in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to [AIEEE-2006]
(A) 300 N (B) 150 N (C) 3 N (D) 30 N

10. A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m which
applying the force and the ball goes upto 2m height further, find the magnitude of the force.
Consider g = 10 m/s2 [AIEEE-2006]
(A) 22 N (B) 4 N (C) 16 N (D) 20 N

11. A wire elongates by l mm when a load W is hanged from it. If the wire goes over a pulley and two weights W
each are hung at the two ends, the elongation of the wire will be (in mm) [AIEEE-2006]

l
(A) (B) l (C) 2l (D) zero
2

12. A block of mass ‘m’ is connected to another block of mass ‘M’ by a spring (massless) of spring constant ‘k’.
The blocks are kept on a smooth horizontal plane. Initially the blocks are at rest and the spring is unstretched.
Then a constant force ‘F’ starts acting on the block of mass ‘M’ to pull it. Find the initial force on the block
of mass ‘m’. [AIEEE-2007]

mF (M + m)F MF
(A) (B) (C) 0 (D)
M m (m + M)

13. A particle of mass m is at rest at the origin at time t = 0. It is subjected to a force F(t) = Foe–bt in the x
direction. Its speed n(t) is depicted by which of the following curves ? [AIEEE-2012]

(A) (B) (C) (D)

14. If a body loses half of its velocity on penetrating 3 cm in a wooden block, then how much will it penetrate
more before coming to rest? [AIEEE-2002]
(A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm

15. A spring of force constant k is cut into two pieces such that one piece is double the length of the other.
Then the long piece will have a force constant of
(A) (2/3) k (B) (3/2) k (C) 3k (D) 6k [JEE 1999]
16. In the figure masses m 1, m 2 and M are 20 kg, 5 kg and 50 kg respectively.
The co-efficient of friction between M and ground is zero. The co-efficient
of friction between m 1 and M and that between m2 and ground is 0.3. The
pulleys and the string are massless . The string is perfectly horizontal
between P1 and m 1 and also between P2 and m 2 . The string is perfectly
vertical between P 1 and P2.An external
horizontal force F is applied to the mass M. Take g = 10 m/s 2.
(a) Draw a free-body diagram for mass M, clearly showing all the forces.
(b) Let the magnitude of the force of friction between m 1 and M be f 1 and that between m2 and ground be f 2.
For a particular F it is found that f 1 = 2 f2 . Find f 1 and f 2 . Write down equations of motion of all the
masses . Find F, tension in the string and accelerations of the masses. [JEE 2000]

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [11]
17. The pulleys and strings shown in the figure are smooth and of
negligible mass. For the system to remain in equilibrium, the
angle q should be [JEE (Scr) 2001]
(A) 0° (B) 30°
(C) 45° (D) 60°

18. 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 [JEE (Scr) 2001]

(A) 2 Mg (B) 2 mg
(C) ( M + m) 2 + m 2 g (D) (M + m) 2 + M 2 g

19. A block of mass 3 kg is placed on a rough horizontal surface whose coefficient

of friction is 1 2 3 minimum value of force F (shown in figure) for which the block
starts to slide on the surface. (g=10m/s2)
(A) 20 N (B) 20 3 N

(C) 10 3 N (D) None of these [JEE (Scr) 2003]

20. Two blocks A and B of equal masses are released from an inclined plane of
inclination 45° at t = 0. Both the blocks are initially at rest. The coefficient
of kinetic friction between the block A and the inclined plane is 0.2 while it
is 0.3 for block B. Initially, the block A is 2 m behind the block B. When
and where their front faces will come in a line.
[Take g = 10 m/s2].
[JEE 2004]

21. Two blocks A and B of masses 2m and m, respectively, are connected by a massless and
inextensible string.The whole system is suspended by a massless spring as shown in the
figure.The magnitudes of acceleration of A and B, immediately after the string is cut, are
respectively [JEE 2006]
(A) g, g (B) g, g/2
(C) g/2, g (D) g/2, g/2

22. A circular disc with a groove along its diameter is placed horizontally. A block
of mass 1 kg is placed as shown. The co-efficient of friction between the block
and all surfaces of groove in contact is m = 2/5. The disc has an acceleration
of 25 m/s2. Find the acceleration of the block with respect to
disc. [JEE 2006]

23. Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on
a frictionless horizontal surface with the string held tight so that each mass is at a distance ‘a’ from the
center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but
constant force F. As a result, the particles move towards each other on the surfaces. The magnitude of
acceleration, when the separation between them becomes 2x, is [JEE 2007]
F a F x
(A) 2m a 2 - x 2 (B) 2m a 2 - x 2

F x F a2 - x2
(C) (D)
2m a 2m x

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [12]
24. Statement-1 : A cloth covers a table. Some dishes are kept on it. The cloth can be pulled out without
dislodging the dishes from the table
because
Statement-2 : For every action there is an equal and opposite reaction
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True [JEE 2007]
25. Statement-1 : It is easier to pull a heavy object that to push it on a level ground. [JEE 2008]
and
Statement-2 : The magnitude of frictional force depends on the nature of the two surfaces in contact.
(A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
26. A piece of wire is bent in the shape of a parabola y = kx2 (y-axis vertical) with a bead of mass m on it. The
bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at
lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with
a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay
at rest with respect to the wire, from the y-axis is [JEE 2009]
a a 2a a
(A) gk (B) 2gk (C) gk (D) 4gk

27. A block of mass m is on an inclined plane of angle q. The coefficient of friction between the
block and the plane is m and tan q > m. The block is held stationary by applying a force P
parallel to the plane. The direction of force pointing up the plane is taken to be positive. As
P is varied from P1 = mg(sinq – m cosq) to P2 = mg(sinq + m cosq), the frictional force
f versus P graph will look like

(A) (B) (C) (D) [JEE 2010]

28. A block of mass 2 kg is free to move along the x-axis. It is at rest and from
t = 0 onwards it is subjected to a time-dependent force F(t) in the x direction.
The force F(t) varies with t as shown in the figure. The kinetic energy of the
block after 4.5 seconds is
(A) 4.50 J (B) 7.50 J
(C) 5.06 J (D) 14.06 J [JEE 2010]
29. A block is moving on an inclined plane making and angle 45º with the horizontal and the coefficient of friction
is m. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from
sliding down. If we defined N = 10m, then N is : [JEE 2011]

30. A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle q with the horizontal.
A horizontal force force of 1 N acts on the block through its centre of mass as shown in the figure. The block
remains stationary if (take g = 10 m/s2) [JEE 2012]
(A) q = 45º
(B) q > 45º and a frictional force acts on the block towards P
(C) q > 45º and a frictional force acts on the block towards Q
(D) q < 45 and a frictional force acts on the block towards Q

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [13]
 x3
31. A block of mass m is placed on a surface with a vertical cross section given by y = . If the coefficient of
6
friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is :

2 1 1 1
(A) m (B) m (C) m (D) m [IIT Mains - 2014]
3 3 2 6

32. A block of mass m1 = 1 kg another mass m2 = 2 kg, are placed together (see figure) on an nclined plane with
angle of inclination q. Various values of q are given in List-I. The coefficient of friction between the block m1
and the plane is always zero. The coefficient of static and dynamic friction between the block m2 and the
plane are equal to m = 0.3. In List-II expressions for the friction on block m2 are given. Match the correct
expression of the friction in List-II with the angles given in List-I, and choose the correct option. The acceleration
due to gravity is denoted by g.
(Useful information : tan (5.5º) » 0.1 ; tan(11.5º) » 0.2 ; tan(16.5º) » 0.3] [JEE Advance - 2014]
List-I List-II
(P) q = 5º (1) m2g sinq
(Q) q = 10º (2) (m1 + m2)g sinq
(R) q = 15º (3) mm2g cosq
(S) q = 20º (4) m(m1 + m2)g cosq
Code :
(A) P-1, Q-1, R-1, S-3 (B) P-2, Q-2, R-2, S-3
(C) P-2, Q-2, R-2, S-4 (D) P-2, Q-2, R-3, S-3

33. [IIT Mains - 2015]

Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed
(held at rest) against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and
between block B and the wall is 0.15, the frictional force applied by the wall on block B is :
(A) 150 N (B) 100 N (C) 80 N (D) 120 N

34. A point particle of mass m, moves along the uniformly rough track PQR as shown in the figure. The coefficient
of friction, between the particle and the rough track equals m. The particle is released from rest, from the point
P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track,
are equal to each other, and no energy is lost when particle changes direction from PQ to QR. The values of
the coefficient of friction m and the distance x(= QR), are, respectively close to

[IIT Mains - 2016]

(A) 0.2 and 6.5 m (B) 0.2 and 3.5 m (C) 0.29 and 3.5 m (D) 0.29 and 6.5 m
Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [14]
35. Two masses m1 = 5 kg and m2 = 10 kg, connected by an inextensible string over a frictionless pulley, are
moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15. The minimum weight m
that should be put on top of m2 to stop the motion is : [IIT Main-2018]

(A) 10.3 kg (B) 18.3 kg (C) 27.3 kg (D) 43.3 kg

36. A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m = 0.4 kg is at rest
on this surface. An impulse of 1.0 N s is applied to the block at time t = 0 so that it starts moving along the
x-axis with a velocity v(t) = v0e–t/t, where v0 is a constant and t = 4 s. The displacement of the block, in
metres, at t = t is _________. Take e–1 = 0.37. [JEE-Advanced-2018]

37. A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two
other blocks of mass M and 2M using two massless pulleys and strings. The accelerations of the blocks are
a1, a2 and a3 as shown in the figure. The system is released from rest with the spring in its unstretched state.
The maximum extension of the spring is x0. Which of the following option(s) is/are correct ?
[g is acceleration due to gravity. Neglect friction] [JEE-Advanced-2019]

x0
(A) At an extension of of the spring, the magnitude of acceleration of the block connected to the spring
4
3g
is
10
x0
(B) When spring achieves an extension of for the first time, the speed of the block connected to the
2
M
spring is 3g
5k
4Mg
(C) a2 – a1 = a1 – a3 (D) x0 =
k
38. Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the
right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the
left finger slips with respect to the scale and the right finger does not. After some distance, the left finger
stops and the right one starts slipping. Then the right finger stops at a distance xR from the center (50.00 cm)
of the scale and the left one starts slipping again. This happens because of the difference in the frictional
forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale
are 0.40 and 0.32, respectively, the value of xR (in cm) is ______. [JEE-Advanced-2020]
Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [15]
 HINTS FOR DIFFICULT PROBLEMS OF EX-II
3. See motion of man from rope frame (NIF) and motion of A from ground frame.

4. m1 and 5 kg form a simple atwood machine.

5. Treat (M + m) as a system.

6. Find net pulling force on system and compare with fmax.

8. Till B hits floor, A and B move under tension & gravity. After that A moves only under gravity till string
is tight again. (i.e. position of start of slack)

10. Till 1 sec, f ® fs, f = fmax at t = 1and f = fk from t = 1 to t = 3.

ANSWER KEY
EXERCISE-I
1. C 2. B 3. C 4. C 5. B 6. A 7. A
8. C 9. B 10. C 11. A 12. D 13. B 14. B
15. A 16. C 17. A 18. A 19. B
20. (A) ® S ; (B) ® Q ; (C) ® P ; (D) ® R

EXERCISE-II
4g 13mg
1. 3:1 2. 240 N 3. a= ,T= 4. 10/3 kg
9 18
5. 3/4 6. 0, 10N ˆi 7. 30 N 8. (a) 2 ms–2, (b) 2.4 N, 0.3 (c) 0.2
s
9. 0.5 sec 10. ms = 0.4 , mk = 0.3

EXERCISE-III
1. A 2. B 3. D 4. B 5. C 6. C 7. A
8. A 9. D 10. A 11. B 12. C 13. B 14. A
2
15. B 16. (b) a = 3/5 m/s , T = 18 N, F = 60N
17. C 18. D 19. A 20. 11.313 m 21. C 22. 10 m/s2 23. B
24. B 25. B 26. B 27. A 28. C 29. 5 30. AC
31. D 32. D 33. D 34. C 35. C 36. 6.30 37. C
38. 25.60

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [16]
 CIRCULAR MOTION & WORK POWER ENERGY
1. A body moving with constant speed in a circular path is continuously accelerated towards the centre of
rotation. The magnitude of this normal acceleration is given by
v2
an = = w2 r
r
where v is the constant speed (v = wr) and
r is the radius of the circular path
dv
Tangential area : at = , a= a 2t + an2
dt

v2
2. Radius of curvature : r =
an

3. According to Newton’s second law, a body moving in a circular path with constant speed must be acted upon
by an unbalanced force which is always directed towards the centre. This necessary unbalanced force is
called the centripetal force.
mv 2
F= = mw2r
r
4. Centrifugal force is a pseudo force which is observed an observer in rotating frame.
r 2 r
Fcf = mwframe r

Work (W) :
The work W done by a constant force F when its point of application
undergoes a displacement s is defined as
W = F.s = Fs cos q
where q is the angle between F and s.Work is a scalar quantity and its SI
units is N-m or joule (J).

Note: Only the component (F cos q) of the force F which is along the displacement contributes to the work done.

If F= Fx î + Fy ˆj + Fz k̂ and s = Dx î + Dyĵ + Dzk̂

r r
then W = F ·s = FxDx + FyDy + Fz D z

5. Work done by a Variable Force : When the magnitude and direction of a force varies with position, The
work done by such a force for an infinitesimal displacement ds is given by
r r
dW = F · d s

In terms of rectangular components,

XB YB ZB
WAB = ò Fx dx + ò Fydy + ò Fz dz
XA YA ZA

6. Work Done by a Spring Force : The work done by the spring force for a displacement from xi to xf is given by

1
Ws = - k x f2 - x i2
2
7. Work Energy theorem :
Work done on a body can produce a change in its kinetic energy. Work is required to produce motion and it
is also required to destroy motion.
W = DK = Kf – Ki

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [17]
8. Conservative Force : The force which does work in complete independence of the path followed the body is
called a conservative force. The gravitational force, spring force and electrostatic force are the examples of
conservative forces.

9. Non-Conservative Force : The work done by a non-conservative force not only depends on the initial and
final positions but also on the path followed. The common examples of such forces are : frictional force and
drag force of fluids.

10. Potential Energy : The potential energy is defined only for conservative forces.
B
UB–UA = – Fc .ds
ò
A

dU
11. Conservative force : Fc = –
dx
dU
At equilibrium, =0
dx
d 2U
The point B is the position of stable equilibrium, because >0
dx 2

d 2U
The point C is the position of unstable equilibrium, because <0
dx 2

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [18]
 EXERCISE-I
1. A man is standing on a rough (m = 0.5) horizontal disc rotating with constant angular velocity of
5 rad/sec. At what distance from centre should he stand so that he does not slip on the disc?
(A) R £ 0.2m (B) R > 0.2 m (C) R > 0.5 m (D) R > 0.3 m
2. A car travelling on a smooth road passes through a curved portion of the road in
form of an arc of circle of radius 10 m. If the mass of car is 500 kg, the reaction on
car at lowest point P where its speed is 20 m/s is
(A) 35 kN (B) 30 kN (C) 25 kN (D) 20 kN

3. A pendulum bob is swinging in a vertical plane such that its angular amplitude is less than 900. At its highest
point, the string is cut. Which trajectory is possible for the bob afterwards.

(A) (B) (C) (D)

4. A conical pendulum is moving in a circle with angular velocity w as shown. If

tension in the string is T, which of following equations are correct ?
(A) T = m w2l (B) T sinq = mw2l
(C) T = mg cosq (D) T = m w2 l sinq

5. A road is banked at an angle of 30° to the horizontal for negotiating a curve of radius 10 3 m. At what
velocity will a car experience no friction while negotiating the curve?
(A) 54 km/hr (B) 72 km/hr (C) 36 km/hr (D) 18 km/hr

6. The ratio of period of oscillation of the conical pendulum to that of the simple pendulum is :
(Assume the strings are of the same length in the two cases and q is the angle made by the string with the
vertical in case of conical pendulum)
(A) cos q (B) cos q (C) 1 (D) none of these

7. A particle is moving in a circle :

(A) The resultant force on the particle must be towards the centre.
(B) The cross product of the tangential acceleration and the angular velocity will be zero.
(C) The direction of the angular acceleration and the angular velocity must be the same.
(D) The resultant force may be towards the centre.

8. Which vector in the figures best represents the acceleration of a pendulum mass at the intermediate point in
its swing?

(A) (B) (C) (D)

9. The dumbell is placed on a frictionless horizontal table. Sphere A is attached to a

frictionless pivot so that B can be made to rotate about A with constant angular
velocity. If B makes one revolution in period P, the tension in the rod is

4p2 Md 8p2 Md 4p 2 Md 2 Md
(A) 2 (B) 2 (C) (D)
P P P P

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [19]
10. A body of mass m accelerates uniformly from rest to a speed v0 in time t0. The work done on the body till any
time t is
æ t2 ö 3
1 ç ÷ 1 æ t0 ö æ t ö æ t ö
2
(A) mv0 ç 2 ÷ (B) mv02 ç ÷ (C) mv02 çç ÷
÷ (D) mv02 çç ÷
÷
2 t
è 0 ø 2 è t ø è t0 ø t
è 0ø

11. F = 2x2 – 3x – 2. Choose correct option

(A) x = – 1/2 is position of stable equilibrium (B) x = 2 is position of stable equilibrium
(C) x = – 1/2 is position of unstable equilibrium (D) x = 2 is position of neutral equilibrium
12. Assume the aerodynamic drag force on a car is proportional to its speed. If the power output from the engine
is doubled, then the maximum speed of the car.
(A) is unchanged (B) increases by a factor of 2
(C) is also doubled (D) increases by a factor of four.

13. A light spring of length 20 cm and force constant 2 N/cm is placed vertically on a table.A small block of mass
1 kg falls on it. The length h from the surface of the table at which the ball will have the maximum velocity is
(A) 20 cm (B) 15 cm (C) 10 cm (D) 5 cm
14. A particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the
circle at a vertical distance h below the highest point, such that
(A) h = R (B) h = R/3 (C) h = R/2 (D) h = 2R
15. A small cube with mass M starts at rest at point 1 at a height 4R, where R is the
radius of the circular part of the track. The cube slides down the frictionless track
and around the loop. The force that the track exerts on the cube at point 2 is nearly
_____ times the cube's weight Mg.
(A) 1 (B) 2 (C) 3 (D) 4

16. A particle is rotated in a vertical circle by connecting it to a light rod of length l and keeping the other end of
the rod fixed. The minimum speed of the particle when the light rod is horizontal for which the particle will
complete the circle is

(A) gl (B) 2gl (C) 3gl (D) none

17. The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular
acceleration constantly decreasing?

(A) (B) (C) (D)

18. A particle is released from rest at origin. It moves under influence of potential field U = x2 – 3x , kinetic energy
at x = 2 is
(A) 2 J (B) 1 J (C) 1.5 J (D) 0 J
19. The work done in joules in increasing the extension of a spring of stiffness 10 N/cm from 4 cm to 6 cm is:
(A) 1 (B) 10 (C) 50 (D) 100
20. When a internal conservative force does positive work within the system
(A) the potential energy increases (B) the potential energy decreases
(C) total energy increases (D) total energy decreases

21. A body of mass 1 kg starts moving from rest at t = 0, in a circular path of radius 8 m. Its kinetic energy varies
as a function of time as : K.E. = 2t2 Joules, where t is in seconds. Then
(A) tangential acceleration = 4 m/s2 (B) power of all forces at t = 2 sec is 8 watt
(C) first round is completed in 2 sec. (D) tangential force at t = 2 sec is 4 newton.

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [20]
Question No. 22 to 23 (2 questions)
A spring block system is placed on a rough horizontal floor. The block is pulled towards right to give spring an
2mmg mmg
elongation less than but more than and released.
K K
22. Which of the following laws/principles of physics can be applied on the spring
block system
(A) conservation of mechanical energy (B) conservation of momentum
(C) work energy principle (D) None
23. The correct statement is
(A) The block will cross the position where spring is in natural length.
(B) The block come to rest when the forces acting on it are exactly balanced
(C) The block will come to rest when the work done by friction becomes equal to the change in energy
stored in spring.
(D) None
Match The Column :
24. Mark matching options for situation shown in respective figures at the instant when particle/car is located on
the x-axis as shown.
Column - I Column - II
(A) Block attached to string is moving (P) Force due to friction may have
along a circle on rough surface. non-zero x-component

(B) Block is placed on a disk rotating with (Q) Force due to friction may have
non-uniform angular velocity as shown non-zero y-component
below. There is no slipping between
block and disk

(C) Car moving on ground along a circular (R) Force due to friction may have
horizontal track at a constant speed non-zero z-component

(D) Car moving on ground at constant speed (S) Force due to friction may be zero.
along a circular banked track

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [21]
 EXERCISE-II
1. Two strings of length l = 0.5 m each are connected to a block of mass m = 2 kg at one
end and their ends are attached to the point A and B 0.5 m apart on a vertical pole which
T1
rotates with a constant angular velocity w = 7 rad/sec. Find the ratio
T2
of tension in the upper string (T1) and the lower string (T2). [Use g = 9.8 m/s2]

2. The P.E. of a particle oscillating on x-axis is given as U = 20 + (x – 2)2 here U is in Joules & x is in meters.
Total mechanical energy of particle is 36 J
(a) Find the mean position
(b) Find the max. K.E. of the particle

3. Consider the shown arrangement when a is bob of mass ‘m’ is suspended by means of
a string connected to peg P. If the bob is given a horizontal velocity
r
u having
magnitude 3gl , find the minimum speed of the bob in subsequent motion.

4. A block of mass m placed on a smooth horizontal surface is attached to a spring

and is held at rest by a force P as shown. Suddenly the force P changes its
direction opposite to the previous one. How many times is the maximum extension
l2 of the spring longer compared to its initial compression l1?

5. A particle is confined to move along the +x axis under the action of a force F(x) that
is derivable from the potential U(x) =ax3-bx.
(a) Find the expression for F(x)
(b) When the total energy of the particle is zero, the particle can be trapped with in
the interval x=0 to x=x1. For this case find the values of x1.
(c) Determine the maximum kinetic energy that the trapped particle has in its motion. Express all answers in
terms a and b.

6. Two blocks of mass m1 = 10 kg and m2 = 5kg connected to each other by a massless inextensible string of
length 0.3m are placed along a diameter of a turn table. The coefficient of friction between the table and m1
is 0.5 while there is no friction between m2 and the table. The table is rotating with an angular velocity of
10rad/sec about a vertical axis passing through its centre. The masses are placed along the diameter of the
table on either side of the centre O such that m1 is at a distance of 0.124m from O. The masses are observed
to be at rest with respect to an observer on the turn table.
(i) Calculate the frictional force on m1
(ii) What should be the minimum angular speed of the turn table so that the masses will slip from this position.
(iii) How should the masses be placed with the string remaining taut, so that there is no frictional force acting
on the mass m 1.

7. The ends of spring are attached to blocks of mass 3kg and 2kg. The 3kg block rests on a
horizontal surface and the 2kg block which is vertically above it is in equilibrium producing
a compression of 1cm of the spring. The 2kg mass must be compressed further by at least
_______, so that when it is released, the 3 kg block may be lifted off
the ground.

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [22]
 EXERCISE-III
1. The maximum velocity (in ms–1) with which a car driver must traverse a flat curve of radius 150 m and
coefficient of friction 0.6 to avoid skidding is [AIEEE-2002]
(A) 60 (B) 30 (C) 15 (D) 25

2. A force F = (5 î + 3 ĵ + 2k̂ )N is applied over a particle which displaces it from origin to the point r = ( 2 î – ĵ) m .
The work done on the particle in joules is [AIEEE-2004]
(A) –7 (B) +7 (C) +10 (D) +13

3. A body of mass m, accelerates uniformly from rest to v1 in time t1. The instantaneous power delivered to the
body as a function of time t is [AIEEE-2004]

mv 1t mv 12 t mv 1t 2 mv 12 t
(A) (B) (C) (D)
t1 t12 t1 t1

4. 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. If follows that [AIEEE-2004]
(A) its velocity is constant (B) its acceleration is constant
(C) its kinetic energy is constant (D) it moves in a straight line

5. A light wire fixed at the upper end stretches by length l by applying a force F. The work done in stretching is

F Fl
(A) (B) Fl (C) 2Fl (D) [AIEEE-2004]
2l 2
6. A particle of mass 100 g is thrown vertically upwards with a speed of 5 m/s, the work done by the force of
gravity during the time the particle goes up is [AIEEE-2006]
(A) 0.5 J (B) –0.5 J (C) –1.25 J (D) 1.25 J

æ x4 x2 ö
7. The potential energy of a 1 kg particle free to move along the x-axis is given by V( x ) = ç – ÷ J . The total
ç 4 2 ÷ø
è
mechanical energy of the particle 2J. Then, the maximum speed (in m/s) is [AIEEE-2006]

3 1
(A) 2 (B) (C) 2 (D)
2 2

8. A mass of M kg is suspended by a weightless string. The constant horizontal force that is required to
displace it until the string makes an maximum of 45º with the initial vertical direction is [AIEEE-2006]

Mg
(A) Mg( 2 – 1) (B) Mg( 2 + 1) (C) Mg 2 (D)
2

9. A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes a uncompressed spring, and
compresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is
10,000. N/m. The spring compresses by [AIEEE-2007]
(A) 5.5 cm (B) 2.5 cm (C) 11.0 cm (D) 8.5 cm

10. An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to
be in the range [AIEEE-2008]
(A) 200 J – 500 J (B) 2 × 105J – 3 × 105J (C) 20,000J – 50, 000J (D) 2,000 J – 5,000 J

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [23]
 r
11. A force F = - K ( y î + xĵ) where K is a positive constant, acts on a particle moving in the x-y 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
r
to the pint (a,a). The total work done by the force F on the particle is [JEE 98]
2 2 2 2
(A) – 2Ka (B) 2Ka (C) – Ka (D) Ka

12. A stone is tied to a string of length l is whirled in a vertical circle with the other end of the string at the centre.
At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the
change in its velocity at it reaches a position where the string is horizontal is [JEE 98]
(A) (u 2 - 2 gl ) (B) 2gl (C) ( u 2 - gl ) (D) 2( u 2 - gl )

13. A particle is suspended vertically from a point O by an inextensible massless

string of length L. A vertical line AB is at a distance L/8 from O as shown. The
object given a horizontal velocity u. At some point, its motion ceases to be
circular and eventually the object passes through the line AB. At the
instant of crossing AB, its velocity is horizontal. Find u. [JEE 99]

14. A long horizontal rod has a bead which can slide along its length, and initially placed at a distance L from
one end of A of the rod. The rod is set in angular motion about A with constant angular acceleration a. If the
coefficient of friction between the rod and the bead is m and gravity is neglected, then the time after which
the bead starts slipping is [JEE 2000]

m m 1
(A) (B) (C) (D) infinitesimal
a a ma

15. A small block is shot into each of the four tracks as shown below. Each of the tracks risks to the same
height. The speed with which the block enters the track is the same in all cases. At the highest point of the
track, the normal reaction is maximum in [JEE(Scr)’2001]

(A) (B) (C) (D)

16. An insect crawls up a hemispherical surface very slowly (see the figure). 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 a with the
vertical, the maximum possible value of a is given by [JEE(Scr.)’2001]
(A) cot a=3 (B) tan a = 3 (C) sec a=3 (D) cosec a=3

17. A small ball of mass 2 × 10–3 Kg having a charge of 1 mc is suspended by a string of length 0.8m. Another identical
ball having the same charge is kept at the point of suspension. Determine the minimum horizontal velocity which
should be imparted to the lower ball so that it can make complete revolution. [JEE’2001]

18. A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum,
r
its acceleration vector a is correctly shown in [JEE (Scr.)’2002]

(A) (B) (C) (D)

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [24]
19. A particle, which is constrained to move along the x-axis, is subjected to a force in the same direction which
varies with the distance x of the particle x of the particle from the origin as F(x) = – kx + ax2. Here k and a are
positive constants. For x ³ 0, the functional form of the potential energy U (x) of the particle is
[JEE (Scr.)’2002]

(A) (B) (C) (D)

20. 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 unstretched. Then the maximum extension in the spring is
[JEE (Scr.)’2002]
(A) 4 Mg/k (B) 2 Mg/k (C) Mg/k (D) Mg/2k

21. A spherical ball of mass m is kept at the highest point in the space between two fixed,
concentric spheres A and B (see figure). The smaller sphere A has a radius R and the
space between the two spheres has a width d. The ball has a diameter very slightly
less than d. All surfaces are frictionless. The ball is given a gentle push (towards the
right in the figure). The angle made by the radius vector of the ball with
the upward vertical is denoted by q (shown in the figure). [JEE' 2002]
(a) Express the total normal reaction force exerted by the spheres on the ball as a function of angle q.
(b) Let NA and NB denote the magnitudes of the normal reaction force on the ball exerted by the spheres A and
B, respectively. Sketch the variations of NA and NB as functions of cosq in the range 0 £ q £ p by drawing two
separate graphs in your answer book, taking cosq on the horizontal axes.

22. In a region of only gravitational field of mass 'M' a particle is shifted from A
to B via three different paths in the figure. The work done in different paths
are W 1, W 2, W 3 respectively then [JEE (Scr.)’2003]
(A) W 1 = W 2 = W 3 (B) W 1 = W 2 > W 3
(C) W 1 > W 2 > W 3 (D) W 1 < W 2 < W 3

23. A particle is placed at the origin and a force F = kx is acting on it (where k is a positive constant). If
U(0) = 0, the graph of U(x) versus x will be (where U is the potential energy function) [JEE' 2004(Scr)]

(A) (B) (C) (D)

24. STATEMENT-1 : A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due
to friction between the block and the surface after moving through a certain distance. The surface is now
tilted to an angle of 30° with the horizontal and the same block is made to go up on the surface with the same
initial velocity v. The decrease in the mechanical energy in the second situation is smaller than that in the
first situation. [JEE 2007]
STATEMENT-2 : The coefficient of friction between the block and the surface decreases with the increase in
the angle of inclination.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [25]
25. A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just
sufficient to make it reach the point B. The angle q at which the speed of the bob is half of that at A, satisfies
[JEE 2008]

p p p
(A) q = (B) <q<
4 4 2
p 3p 3p
(C) <q< (D) <q<p
2 4 4
.

26. A light inextensible string that goes over a smooth fixed pulley as shown in the figure
connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done
(in joules) by the string on the block of mass 0.36 kg during the first second after the
system is released from rest. [JEE 2009]

27. A ball of mass (m) 0.5 kg is attached to the end of a string having length (L)
0.5 m. The ball is rotated on a horizontal circular path about vertical axis.
The maximum tension that the string can bear is 324 N. The maximum
possible value of angular velocity of ball (in radian/s) is : [JEE 2011]
(A) 9 (B) 18
(C) 27 (D) 36

28. A block of mass 0.18 kg is attached to a spring of force-constant 2 N/m. The coefficient of friction between
the block and the floor is 0.1. Initially the block is at rest and the spring is un-stretched. An impulse is given
to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time.
The initial velocity of the block in m/s is V = N/10. [JEE 2011]
Then N is

29. Consider a disc rotating in the horizontal plane with a constant angular speed w about its centre O. The disc
has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the
figure. When the disc is in the orientation as shown, two pebbles P and Q are simultaneously projected at an
angle towards R. The velocity of projection is in the y-z plane and is same for both pebbles with respect to the
disc. Assume that (i) they land back on the disc before the disc has completed 1/8 rotation, (ii) their range is
less than half the disc radius, and (iii) w remains constant throughout. Then [JEE 2012]
(A) P lands in the shaded region and Q in the unshaded region.
(B) P lands in the unshaded region and Q in the shaded region.
(C) Both P and Q land in the unshaded region.
(D) Both P and Q land in the shaded region.

30. A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to
the particle. If the initial speed (in ms–1) of the particle is zero, the speed (in ms–1) after 5 s is
[JEE Adv. -2013]

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [26]
Paragraph for Questions 31 and 32 [JEE Adv. -2013]
A small block of mass 1 kg is released from rest at the
top of a rough track. The track is a circular arc of radius
40 m. The block slides along the track without toppling
and a frictional force acts on it in the direction opposite to
the instantaneous velocity. The work done in overcoming
the friction up to the point Q, as shown in the figure below,
is 150 J.
(Take the acceleration due to gravity, g = 10 ms–2).

31. The magnitude of the normal reaction that acts on the block at the point Q is
(A) 7.5 N (B) 8.6 N (C) 11.5 N (D) 22.5 N

32. The speed of the block when it reaches the point Q is

(A) 5 ms–1 (B) 10 ms–1 (C) 10 3 ms –1 (D) 20 ms–1

33. When a rubber-band is stretched by a distance x, it exerts a restoring force of magnitude F = ax + bx2 where
a and b are constants. The work done in stretching the unstretched rubber-band by L is :[IIT Mains - 2014]

1 2 3 aL2 bL3 1 æç aL2 bL3 ö÷

(A) (aL + bL ) (B) + (C) 2 ç 2 + 3 ÷ (D) aL2 + bL3
2 2 3 è ø

34. Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1
kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure
given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n × 10) Joules.
The value of n is (take acceleration due to gravity = 10 ms–2) [JEE Advance - 2014]

35. A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is
fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it
slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is
(A) Always radially outwards. [JEE Advance - 2014]
(B) Always radially inwards.
(C) Radially outwards initially and radially inwards later.
(D) Radially inwards initially and radially outwards later.

36. Distance of the centre of mass of a solid uniform cone from its vertex is z0. If the radius of its base is R and
its height is h then z0 is equal to : [JEE Main - 2015]

3h2 h2 3h 5h
(A) (B) (C) (D)
8R 4R 4 8

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [27]
37. A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1m 1000 times. Assume
that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up
considering the work done only when the weight is lifted up? Fat supplies 3.8 × 107 J of energy per kg which
is converted to mechanical energy with a 20% efficiency rate. Take : g = 9.8 ms–2. [JEE Main - 2016]
(A) 2.45 × 10–3 kg (B) 6.45 × 10–3 kg (C) 9.89 × 10–3 kg (D) 12.89 × 10–3 kg

38. A body of mass m = 10–2 kg is moving in a medium and experiences a frictional force F = –kv2. Its initial
1
speed is v0 = 10 ms–1. If, after 10 s, its energy is mv 20 , the value of k will be : [JEE Main - 2017]
8
(A) 10–3 kg s–1 (B) 10–4 kg m–1 (C) 10–1 kg m –1 s–1 (D) 10–3 kg m–1

39. A time dependent force F = 6t acts on a particle of mass 1 kg. If the particle starts from rest, the work done
by the force during the first 1 sec. will be [JEE Main - 2017]
(A) 22 J (B) 9 J (C) 18 J (D) 4.5 J

40. A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional
to the nth power of R. If the period of rotation of the particle is T, then : [IIT Main-2018]
n
+1
(A) T µ Rn/2 (B) T µ R3/2 for any n (C) T µ R 2 (D) T µ R(n+1)/2

k
41. A particle is moving in a circular path of radius a under the action of an attractive potential U = – . Its total
2r 2
energy is : [IIT Main-2018]

3 k k k
(A) – (B) – (C) (D) Zero
2 a2 4a 2 2a2
42. A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-
axis. Its kinetic energy K changes with time as dK/dt = gt, where g is a positive constant of appropriate
dimensions. Which of the following statements is (are) true? [JEE-Advanced-2018]
(A) The force applid on the particle is constant.
(B) The speed of the particle is proportional to time.
(C) The distance of the particle from the origin incrfeases linearly with time.
(D) The force is conservative.

43. A ball is projected from the ground at an angle of 45º with the horizontal surface. It reaches a maximum
height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic
energy. Immediately after the bounce, the velocity of the ball makes an angle of 30º with the horizontal
surface. The maximum height it reaches after the bounce, in metres, is _____. [JEE-Advanced-2018]

44. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force
r
F = (ayi$ + 2ax$j)N . Where x and y are in meter and a = –1 Nm –1. The work done on the particle by this force
r
F will be ..... Joule. [JEE-Advanced-2019]

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [28]
 Hints for Difficult Problems of Advance-II
Exercise-II
2. At mean position U is minimum and KE is maximum.
3. It perform circular motion upto some angle and after that it will perform projectile motion.
6. Use centrifugal force
Exercise-III
13. It perform circular motion upto some angle and after that it will perform projectile motion.
14. Use required friction = available friction
16. Angle of repose
21. Net force = ma
28. Apply work energy theorem

ANSWER KEY
EXERCISE-I
1. A 2. C 3. C 4. A 5. C 6. B 7. D
8. B 9. B 10. A 11. A 12. B 13. B 14. B
15. C 16. B 17. A 18. A 19. A 20. B 21. B
22. C 23. C
24. (A) ® Q ; (B) ® PQ ; (C) ® P ; (D) ® PRS

EXERCISE-II
1 gl
1. 9 2. (a) x = 2, (b) 16 J 3. 4. 3
3 3

b 2b b
5. F = -3ax2 + b, x = , KEmax =
a 3 3 a
6. (i) 36N, (ii) 11.66rad/sec ,(iii) 0.1m, 0.2m 7. 2.5cm

EXERCISE-III
1. B 2. B 3. B 4. C 5. D 6. C 7. B

æ3 3 ö
8. A 9. A 10. D 11. C 12. D 13. u= gL çç + 2 ÷÷
è 2 ø
14. A 15. A 16. A 17. 5.79 m/s 18. C 19. D 20. B

21. (a) N=3mg cosq – 2mg, (b) 22. A

10
23. A 24. C 25. D 26. Work = T.S. = 4.8 × = 8N 27. D
6
28. 4 29. C 30. 5 31. A 32. B 33. B 34. 5
35. D 36. C 37. D 38. B 39. D 40. D 41. D
42. A,B,D 43. 30.00 44. 0.75

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [29]
 CENTRE OF MASS MOMENTUM & COLLISION
The action of force with respect to time is defined in terms of Impulse, that is,

I= ò Fdt = mv – mv =Dp
f i

In the absence of a net external force, the momentum of a system is conserved.

dP
i.e. = Fext = 0
dt
p = p1 + p2 + ............+ pN = constant
1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a
very short time interval.

2. Types of collision: Elastic and Inelastic

Collisions may be either elastic or inelastic. Linear momentum is conserved in both cases.
(i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved.
(ii) In an inelastic collision, the total kinetic energy of the system changes.
(iii) In a completely inelastic collision, the two bodies couple or stick togehter.

3. Coefficient of Restitution : It is defined as the ratio of the velocity of separation to the velocity of approach
of the two colliding bodies.

rel. velocity of separation

e=
rel. velocity of approach

For a perfectly elastic collision, e = 1

For an inelastic collision, 0 < e < 1
For completely inelastic collision, e = 0
Note that the velocity of approach and the velocity of separation are always taken along the normal to the
striking surface.

CENTRE OF MASS
1. Discrete System : The position vector of the centre of mass is

m1r1 + m 2 r2 + ......... + m n rn
rc =
m1 + m 2 + .........m n
r r r
where r1, r2 ,..., rn are the position vectors of masses m1, m 2, ...., mn respectively..
The components of the position vector of centre of mass are defined as

xc =
å mi x i ; yc =
å mi yi ; zc =
å mi zi
M M M
2. Continuous system : The centre of mass of a continuous body is defined as
r 1
rc = ò r dm
M
In the component form
1 1 1
Mò Mò Mò
xc = x dm ; yc = y dm ; zc = z dm

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [30]
3. Centre of Mass of Some Common Systems :
(i) A system of two point masses.
The centre of mass lie closer to the heavier mass.

(ii) A circular cone

h
yc =
4
(iii) A semi-circular ring
2R
yc = ; xc = 0
p
(iv) A semi-circular disc
4R
yc = ; xc = 0
3p
(v) A hemispherical shell
R
yc = ; xc = 0
2
(vi) A solid hemisphere
3R
yc = ; xc = 0
8
4. Motion of the centre of mass :

(i) Velocity : The instantaneous velocity of the centre of mass is defined as

vc =
å mi vi
M
(ii) Acceleration : The acceleration of the centre of mass is defined as

ac =
å mi a i
M
(iii) Momentum : The total momentum of a system of particles is
p = Mvc
(iv) Kinetic Energy : The kinetic energy of a system of particles consisits of two parts.
K = Kc + K’
1
where Kc = Mv c2 , kinetic energy due to motion of c.m. relative to the fixed origin O,
2
1
and K’ = å 2 mi vi2 , kinetic energy of the particles relative to the c.m.
Note that the term K’ may involve translational, rotational or vibrational energies relative to the centre of
mass.
5. Newon’s Laws of a system of particles : The first and second laws of motion for a system of particles are
modified as :
First law : The centre of mass of an isolated system is at rest or moves with constant velocity.
Second law : The net external force acting on a system of total of mass M is related to the acceleration of
centre of mass of the system.
r r
å ext = M a cm
F

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [31]
 EXERCISE-I
ONLY ONE OPTION IS CORRECT
1. A uniform wire of length l is bent into the shape of ‘V’ as shown. The distance of its centre of mass from the
vertex A is

k 3
(A) l/2 (B)
4
k 3
(C) (D) none of these
8
2. Four cubes of side “a” each of mass 40gm, 20gm,10gm and 20gm are arranged in XY
plane as shown in figure. The coordinates of centre of mass of the combination with
respect to O, are :
(A) 19a/18, 17a/18 (B) 17a/18, 11a/18
(C) 17a/18, 13a/18 (D) 13a/18, 17a/18
3. The centre of mass of a non uniform rod of length L whose mass per unit length varies as
r = kx2/L (where k is a constant and x is the distance measured form one end) is at the following distance
from the same end.
(A) 3L/4 (B) L/4 (C) 2L/3 (D) L/3
4. Four particles are in x-y plane at
(1) 1 kg at (0, 0) (2) 2 kg at (1, 0) (3) 3 kg at (1, 2) (4) 4 kg at (2, 0)
The centre of mass is located at
(A) (0.3, 1.2) (B) (1.3, 0.6) (C) (0.5, 1.4) (D) (1.2, 0.3)

5. A large wedge rests on a horizontal frictionless surface, as shown. A block starts from rest
and slides down the inclined surface of the wedge, which is rough. During the motion
of the block, the center of mass of the block and wedge system
(A) does not move (B) moves vertically with increasing speed
(C) moves horizontally with constant speed (D) moves both horizontally and vertically

6. A uniformly thick plate in the shape of an arrowhead has dimensions as shown. The centre of mass lies at a
point

(A) 1.5 cm to the right of O (B) 3 cm to the right of O

(C) O itself (D) 1 cm to the right of O
7. A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless surface. The man
M
walks to other end of the plank. If the mass of the plank is , then the distance that the man moves relative
3
to ground is :
3L L 4L L
(A) (B) (C) (D)
4 4 5 3

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [32]
8. A man weighing 80 kg is standing at the centre of a flat boat and he is 20 m from the shore. He walks 8 m on
the boat towards the shore and then halts. The boat weight 200 kg. How far is he from the shore at the end
of this time ?
(A) 11.2 m (B) 13.8 m (C) 14.3 m (D) 15.4 m

9. A system of N particles is free from any external forces. Which of the following is true for the magnitude of the
total momentum of the system?
(A) It must be zero
(B) It could be non–zero, but it must be constant
(C) It could be non–zero, and it might not be constant
(D) The answer depends on the nature of the internal forces in the system

10. Which of the following must be true for the sum of the magnitudes of the momenta of the individual particles
in the system?
(A) It must be zero
(B) It could be non–zero, but it must be constant
(C) It could be non–zero, and it might not be constant
(D) It could be zero, even if the magnitude of the total momentum is not zero

11. An isolated rail car of mass M is moving along a straight, frictionless track at an initial speed v0. The car is
passing under a bridge when a crate filled with N bowling balls, each of mass m, is dropped from the bridge
into the bed of the rail car. The crate splits open and the bowling balls bounce around inside the rail car, but
none of them fall out.
(a) Is the momentum of the rail car + bowling balls system conserved in this collision?
(A) Yes, the momentum is completely conserved.
(B) Only the momentum component in the vertical direction is conserved.
(C) Only the momentum component parallel to the track is conserved.
(D) No components are conserved.

(b) What is the average speed of the rail car + bowling balls system some time after the collision?
(A) (M + Nm)v0/M (B) Mv0/(Nm + M)
(C) Nmv0/M (D) The speed cannot be determined because there is not enough information
12. A ball strikes a smooth horizontal ground at an angle of 45° with the vertical. What cannot be the possible
angle of its velocity with the vertical after the collision. (Assume e £ 1 ).
(A) 45° (B) 30° (C) 53° (D) 60°
13. Two massless string of length 5 m hang from the ceiling very near to each
other as shown in the figure. Two balls A and B of masses 0.25 kg and 0.5 kg
are attached to the string. The ball A is released from rest at a height 0.45 m
as shown in the figure. The collision between two balls is completely elastic.
Immediately after the collision, the kinetic energy of ball B is 1 J. The velocity
of ball A just after the collision is
(A) 5 ms–1 to the right (B) 5 ms–1 to the left
–1
(C) 1 ms to the right (D) 1 ms–1 to the left
14. Two balls A and B having masses 1 kg and 2 kg, moving with speeds 21 m/s and 4 m/s respectively in
opposite direction, collide head on. After collision A moves with a speed of 1 m/s in the same direction, then
the coefficient of restitution is
(A) 0.1 (B) 0.2 (C) 0.4 (D) None
15. A particle of mass 3m is projected from the ground at some angle with horizontal. The horizontal range is R.
At the highest point of its path it breaks into two pieces m and 2m. The smaller mass comes to rest and
larger mass finally falls at a distance x from the point of projection where x is equal to
3R 3R 5R
(A) (B) (C) (D) 3R
4 2 4

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [33]
16. A small sphere is moving at a constant speed in a vertical circle. Below is a list of quantities that could be
used to describe some aspect of the motion of the sphere.
I – kinetic energy
II – gravitational potential energy
III – momentum
Which of these quantities will change as this sphere moves around the circle?
(A) I and II only (B) I and III only (C) III only (D) II and III only

17. There are some passengers inside a stationary railway compartment. The track is frictionless. The centre of
mass of the compartment itself (without the passengers) is C1, while the centre of mass of the 'compartment
plus passengers' system is C2. If the passengers move about inside the compartment along the track.
(A) both C1 and C2 will move with respect to the ground
(B) neither C1 nor C2 will move with respect to the ground
(C) C1 will move but C2 will be stationary with respect to the ground
(D) C2 will move but C1 will be stationary with respect to the ground

18. Three blocks are initially placed as shown in the figure. Block A has mass m and initial velocity v to the right.
Block B with mass m and block C with mass 4m are both initially at rest. Neglect friction. All collisions are
elastic. The final velocity of block A is
(A) 0.6v to the left (B) 1.4v to the left
(C) v to the left (D) 0.4v to the right

19. A block of mass m starts from rest and slides down a frictionless semi–circular track from a height h as
shown. When it reaches the lowest point of the track, it collides with a stationary piece of putty also having
mass m. If the block and the putty stick together and continue to slide, the maximum height that the
block-putty system could reach is:
(A) h/4
(B) h/2
(C) h
(D) independent of h

20. A boy hits a baseball with a bat and imparts an impulse J to the ball. The boy hits the ball again with the
same force, except that the ball and the bat are in contact for twice the amount of time as in the first hit. The
new impulse equals:
(A) half the original impulse (B) the original impulse
(C) twice the original impulse (D) four times the original impulse

21. Two billiard balls undergo a head-on collision. Ball 1 is twice as heavy as ball 2. Initially, ball 1 moves with a
speed v towards ball 2 which is at rest. Immediately after the collision, ball 1 travels at a speed of v/3 in the
same direction. What type of collision has occured?
(A) inelastic (B) elastic
(C) completely inelastic (D) Cannot be determined from the information given

22. A ball is dropped from a height h. As it bounces off the floor, its speed is 80 percent of what it was just before
it hit the floor. The ball will then rise to a height of most nearly
(A) 0.80 h (B) 0.75 h (C) 0.64 h (D) 0.50 h
23. A ball is projected from ground with a velocity V at an angle q to the vertical. On its path it makes an elastic
collison with a vertical wall and returns to ground. The total time of flight of the ball is
2 v sin q 2v cos q v sin 2q v cos q
(A) (B) (C) (D)
g g g g

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [34]
24. A rocket of mass 4000 kg is set for vertical firing. How much gas must be ejected per second so that the
rocket may have initial upwards acceleration of magnitude 19.6 m/s2. [Exhaust speed of fuel = 980 m/s.]
(A) 240 kg s–1 (B) 60 kg s–1 (C) 120 kg s–1 (D) None

25. A block of mass M is tied to one end of a massless rope. The other end of
the rope is in the hands of a man of mass 2M as shown in the figure. The
block and the man are resting on a rough plank of mass M as
shown in the figure. The whole system is resting on a smooth
horizontal surface. The man pulls th e rope. Pulley is massless and frictionless. What is the displacement of
the plank when the block meets the pulley.(Man does not leave his position on plank during the pull)
(A) 0.5 m (B) 1m (C) zero (D) 2/3 m

ONE OR MORE THAN ONE OPTION MAY BE CORRECT

26. The diagram to the right shows the velocity-time graph for two
masses R and S that collided elastically. Which of the following
statements is true?
(I) R and S moved in the same direction after the collision.
(II) Kinetic energy of the system (R & S) is minimum at t = 2 milli sec.
(III) The mass of R was greater than mass of S.
(A) I only (B) II only (C) I and II only (D) I, II and III

27. An isolated rail car originally moving with speed v0 on a straight, frictionless, level track contains a large
amount of sand. A release valve on the bottom of the car malfunctions, and sand begins to pour out straight
down relative to the rail car.
(a) Is momentum conserved in this process?
(A) The momentum of the rail car alone is conserved
(B) The momentum of the rail car + sand remaining within the car is conserved
(C) The momentum of the rail car + all of the sand, both inside and outside the rail car, is conserved
(D) None of the three previous systems have momentum conservation

(b) What happens to the speed of the rail car as the sand pours out?
(A) The car begins to roll faster
(B) The car maintains the same speed
(C) The car begins to slow down
(D) The problem cannot be solved since momentum is not conserved

28. In an inelastic collision,

(A) the velocity of both the particles may be same after the collision
(B) kinetic energy is not conserved
(C) linear momentum of the system is conserved.
(D) velocity of separation will be less than velocity of approach.
Match The Column :
29. If net external force on a system of particles is zero, then match the following
Column I Column II
(A) Acceleration of centre of mass (P) must be constant
(B) Kinetic energy of the system (Q) must be zero
(C) Velocity of centre of mass (R) may not be zero
(D) Velocity of an individual particle of the system (S) may not be constant

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [35]
 EXERCISE-II
1. Two cars initially at rest are free to move in the x direction. Car A has mass 4 kg and car B has mass 2 kg.
They are tied together, compressing a spring in between them. When the spring holding them together is
burned, car A moves off with a speed of 2 m/s
(a) with what speed does car B leave.
(b) how much energy was stored in the spring before it was burned.

2. A spaceship is moving with constant speed v0 in gravity free space along +Y-axis suddenly shoots out one
third of its part with speed 2v0 along + X-axis. Find the speed of the remaining part.

3. The linear mass density of a ladder of length l increases uniformly from one end A to the other end B,
(a) Form an expression for linear mass density as a function of distance x from end A where linear mass density
l0. The density at one end being twice that of the other end.
(b) find the position of the centre of mass from end A.

4. In the arrangement shown in the figure, mA = 2 kg and mB = 1 kg. String is

light and inextensible. Find the acceleration of centre of mass of
both the blocks. Neglect friction everywhere.

5. A small block of mass 2m initially rests at the bottom of a fixed circular,

vertical track, which has a radius of R. The contact surface between the
mass and the loop is frictionless. A bullet of mass m strikes the block
horizontally with initial speed v0 and remain embedded in the block as the
block and the bullet circle the loop. Determine each of the following in
terms of m, v0, R and g.
(a) The speed of the masses immediately after the impact.
(b) The minimum initial speed of the bullet if the block and the bullet are to successfully execute a complete ride
on the loop.

6. In a game of Carom Board, the Queen (a wooden disc of radius 2 cm and mass 50 gm)
is placed at the exact center of the horizontal board. The striker is a smooth plastic disc
of radius 3 cm and mass 100 gm. The board is frictionless. The striker is given an initial
velocity ‘u’ parallel to the sides BC or AD so that it hits the Queen inelastically with
coefficient of restitution = 2/3. The impact parameter for the collision is ‘d’ (shown in the
figure). The Queen rebounds from the edge AB of the board inelastically with same
coefficient of restitution = 2/3 and enters the hole D following
the dotted path shown. The side of the board is L.
Find the value of impact parameter ‘d’ and the time which the Queen takes to enter hole D after collision with
the striker.

7. A 24–kg projectile is fired at an angle of 53° above the horizontal with an initial speed of 50 m/s. At the
highest point in its trajectory, the projectile explodes into two fragments of equal mass, the first of which falls
vertically with zero initial speed.
(a) How far from the point of firing does the second fragment strike the ground? (Assume the ground is level.)
(b) How much energy was released during the explosion?

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [36]
8. Mass m1 hits & sticks with m2 while sliding horizontally with velocity v along the common line of centres of the
three equal masses (m1 = m2 = m3 = m). Initially masses m2 and m3 are stationary and the spring is unstretched.
Find
(a) the velocities of m 1, m 2 and m 3 immediately after impact.
(b) the maximum kinetic energy of m 3.
(c) the minimum kinetic energy of m 2.
(d) the maximum compression of the spring.

9. Two masses A and B connected with an inextensible string of length l lie on a smooth
horizontal plane. A is given a velocity of v m/s along the ground perpendicular to line AB
as shown in figure. Find the tension in string during their subsequent motion

10. The simple pendulum A of mass mA and length l is suspended from the trolley B of
mass mB. If the system is released from rest at q = 0, determine the velocity
vB of the trolley and tension in the string when q = 90°. Friction is negligible.

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [37]
 EXERCISE-III
1. Two identical particles move towards each other with velocity 2v and v respectively. The velocity of centre of
mass is [AIEEE-2002]

v v
(A) v (B) (C) (D) zero
3 2

2. A machine gun fires a bullet of mass 40 g with a velocity 1200 ms–1. The man holding it can exert a maximum
force of 144 N on the gun. How many bullets can he fire per second at the most? [AIEEE-2004]
(A) One (B) Four (C) Two (D) Three

3. 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) 7.2 J (B) 3.6 J (C) 120 J (D) 1200 J [AIEEE-2004]

4. A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of mass
1/3 M and a body C of mass 2/3 M. The centre of mass of bodies B and C taken together shifts compared to
that of body A towards [AIEEE-2005]
(A) depends on height of breaking (B) does not shift
(C) body C (D) body B
5. The block of mass M moving on the frictionless horizontal surface collides with a spring of spring constant K
and compresses it by length L. The maximum momentum of the block after collision is [AIEEE-2005]

KL2 ML2
(A) MK L (B) (C) zero (D)
2M K

6. A mass ‘m’ moves with a velocity v and collides inelastically with another identical mass at rest. After
collision the 1st mass moves with velocity v / 3 in a direction perpendicular to the initial direction of motion.
Find the speed of the 2nd mass after collision [AIEEE-2005]

(A) v (B) 3 v (C) 2v / 3 (D) v / 3

7. A bomb of mass 16 kg at rest explodes into two pieces of masses of 4 kg and 12 kg. The velocity of the
12 kg mass is 4 ms–1. The kinetic energy of the other mass is [AIEEE-2006]
(A) 96 J (B) 144 J (C) 288 J (D) 192 J

8. Consider a two particle system with particles having masses m1 and m2. If the first particle is pushed
towards the centre of mass through a distance d, by what distance should the second particle be moved, so
as to keep the centre of mass at the same position? [AIEEE-2006]

m2 m1 m1
(A) d (B) m d (C) m + m d (D) m d
1 1 2 2

9. A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms–1. The magnitude of its
momentum is recorded as [AIEEE-2008]
(A) 17.6 kg ms–1 (B) 17.565 kg ms–1 (C) 17.56 kg ms–1 (D) 17.57 kg ms–1

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [38]
10. Statement-1 Two particles moving in the same direction do not lose all their energy in a completely
inelastic collision. [AIEEE-2010]
Statement-2 Principle of conservation of momentum holds true for all kinds of collisions.
(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1
(B) Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1
(C) Statement-1 is true, Statement-2 is false
(D) Statement-1 is false, Statement-2 is true
11. This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the
one that best describes the two Statements :
Statement-1 : A point particle of mass m moving with speed v collides with stationary point particle of mass

1 æ m ö
M. If the maximum energy loss possible is given as f æç mv 2 ö÷ then f = ç ÷.
è2 ø èM+mø

Statement-2 : Maximum energy loss occurs when the particles get stuck together as a result of the collision.
(A) Statement-1 is false, Statement-2 is true.
(B) Statement-1 is true, Statement-2 is true, Statement-2 is the correct explanation of Statement-1.
(C) Statement-1 is true, Statement-2 is true, Statement-2 is not the correct explanation of Statement-1.
(D) Statement-1 is true, Statement-2 is false. [IIT Mains - 2013]

12. Two block of masses 10 kg and 4 kg are connected by a spring of negligible mass and placed on a frictionless
horizontal surface. An impulse gives a velocity of 14 m/s to the heavier block in the direction of the lighter
block. The velocity of the centre of mass is : [IIT (Scr) 2002]
(A) 30 m/s (B) 20 m/s (C) 10 m/s (D) 5 m/s

r r
13. Two balls, having linear momenta p1 = pî and p 2 = - p î , undergo a collision in free space. There is no

r r
external force acting on the balls. Let p'1 and p'2 be their final momenta. The following option(s) is(are) NOT
ALLOWED for any non-zero value of p, a1, a2, b1, b2, c1 and c2. [JEE 2008]
r r r r
(A) p'1 = a1î + b1ˆj + c1k̂ (B) p '1 = c1k̂ (C) p '1 = a 1î + b1ˆj + c1k̂ (D) p'1 = a1î + b1ˆj

r r r r
p'2 = a 2 î + b 2 ˆj p'2 = c 2 k̂ p'2 = a 2 î + b 2 ˆj - c1k̂ p'2 = a 2 î + b1ˆj
Comprehension Q. 14 to Q.16 (3 Questions)
A small block of mass M moves on a frictionless surface
of an inclined plane, as shown in figure. The angle of the
incline suddenly changes from 60° to 30° at point B. The
block is initially at rest at A. Assume that collisions
between the block and the incline are totally inelastic
(g = 10 m/s2).
14. The speed of the block at point B immediately after it
strikes the second incline is :

(A) 60 m/s (B) 45 m/s [JEE 2008]

(C) 30 m/s (D) 15 m/s

15. The speed of the block at point C, immediately before it leaves the second incline is : [JEE 2008]
(A) 120 m/s (B) 105 m/s (C) 90 m/s (D) 75 m/s

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [39]
16. If collision between the block and the incline is completely elastic, then the vertical (upward) component of
the velocity of the block at point B, immediately after it strikes the second incline is : [JEE 2008]

(A) 30 m/s (B) 15 m/s (C) 0 (D) - 15 m/s

17. Two small particles of equal masses start moving in opposite directions from a point A in a
horizontal circular orbit. Their tangential velocities are v and 2v, respectively, as shown in the
figure. Between collisions, the particle move with constant speeds. After making how many
elastic collisions, other than that at A, these two particles will again reach the point A ?
(A) 4 (B) 3 (C) 2 (D) 1 [JEE 2009]

18. Look at the drawing given in the figure which has been drawn with ink of uniform
line-thickness. The mass of ink used to draw each of the two inner circles, and each of
the two line segments is m. The mass of the ink used to draw the outer circle is 6m. The
coordinates of the centres of the different parts are : outer circle (0, 0), left inner circle
(–a, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, –a),
The y-coordinate of the centre of mass of the ink in this drawing is [JEE 2009]
(A) a/10 (B) a/8 (C) a/12 (D) a/3
19. If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame,
one can surely say that [JEE 2009]
(A) linear momentum of the system does not change in time
(B) kinetic energy of the system does not change in time
(C) angular momentum of the system does not change in time
(D) potential energy of the system does not change in time
20. Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses
m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision
with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line.
Find the final speed (in m/s) of the object C. [JEE 2009]

21. A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg
mass reverses its direction and moves with a speed of 2 ms–1. Which of the following statement(s) is (are)
correct for the system of these two masses? [JEE 2010]
(A) Total momentum of the system is 3 kg ms–1 (B) Momentum of 5 kg mass after collision is 4 kg ms–1
(C) Kinetic energy of the centre of mass is 0.75 J (D) Total kinetic energy of the system is 4J

22. A ball of mass 0.2 kg rests on a vertical post of height 5 m. A bullet

of mass 0.01 kg, traveling with a velocity V m/s in a horizontal
direction, hits the centre of the ball. After the collision, the ball and
bullet travel independently. The ball hits the ground at a distance
of 20 m and the bullet at a distance of 100 m from the foot of the
post. The initial velocity V of the bullet is [JEE 2011]
(A) 250 m/s (B) 250 2 m / s
(C) 400 m/s (D) 500 m/s
23. A bob of mass m, suspended by a string of length l1 is given a minimum velocity required to3complete a full
circle in the ertical plane. At the highest point, it collides elastically with another bob of mass m suspended
by a string of length l2, which is initially at rest. Both the strings are mass-less and inextensible. If the
second bob, after collision acquires the minimum speed required to complete a full circle in the vertical
l1
plane, the ratio l is : [JEE Advance 2013]
2

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [40]
24. A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting
the surface. The force on the ball during the collision is proportional to the length of compression of the ball.
Which one of the following sketches describes the variation of its kinetic energy K with time t most
appropriately? The figure are only illustrative and not to the scale [JEE Advance - 2014]

(A) (B) (C) (D)

25. A particle of mass m moving in the x direction with speed 2v is hit by another particle of mass 2m moving in
the y direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the
collision is close to : [JEE Advance - 2015]
(A) 62% (B) 44% (C) 50% (D) 56%

m
26. A particle A of mass m and initial velocity u collides with a particle B of mass which is at rest. The
2
collision is head on, and elastic. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths
lA and lB after the collision is : [IIT Mains - 2017]

lA lA 2 lA 1 lA 1
(A) l = 2 (B) l = 3 (C) l = 2 (D) l = 3
B B B B

27. A block of mass M has a circular cut with a firctionless surface as shown. The block rests on the horizontal
frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system
fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides
down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant,
which of the following options is/are correct ? [JEE Advance - 2017]

2gR
(A) The velocity of the point mass m is : v =
m
1+
M

m
(B) The velocity fo the block M is : V = - 2gR
M
mR
(C) The position of the point mass is : x = - 2
M+m
mR
(D) The x component of displacement of the center of mass of the block M is : -
M+m

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [41]
28. Consider regular polygons with number of sides n = 3,4,5.......... as shown in the figure. The center of mass
of all the polygons is at height h from the ground. They roll on a horizontal surface about the leading vertex
without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass
for each polygon is D. Then D depends on n and h as [JEE Advance - 2017]

æ ö
ç ÷
æ pö æ 2p ö ç 1 ÷ 2æ p ö
(A) D = h sin2 ç ÷ (B) D = h sinç ÷ (C) D = hç - 1 (D) D = h tan ç ÷
ènø è n ø æ ö ÷
p è 2n ø
çç cosç ÷ ÷÷
è ènø ø

29. In a collinear collision, a particle with an initial speed v0 strikes a stationary particle of the same mass. If the
final total kinetic energy is 50% greater than the original kinetic energy, the magnitude of the relative velocity
between the two particles, after collision is : [IIT Main-2018]
v0 v0 v0
(A) (B) (C) 2v 0 (D)
2 4 2

30. A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 Nm–1 and
the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched
condition. Another block of mass 1.0 kg moving with a speed of 2.0 ms–1 collides elastically with the first
block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the
two blocks when the spring returns to its unstretched position for the first time after the collision is _________.
[JEE-Advanced-2018]

31. A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes
elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the
other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from
closed end is L = L0 the particle speed is v = v0. The piston is moved inward at a very low speed V such that
dL
V << v 0 , where dL is the infuitesimal displacement of the piston. Which of the following statement(s) is/
L
are correct ? [JEE-Advanced-2019]

(A) After each collision with the piston, the particle speed increases by 2V
1
(B) The particle’s kinetic energy increases by a factor of 4 when the piston is moved inward from L0 to L
2 0
dL
(C) If the piston moves inward by dL, the particle speed increases by 2v
L
v
(D) The rate at which the particle strikes the piston is
L

Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [42]
 Question Nos. 32 and 33

A projectile is thrown from a point O on the ground at an angle 45° from the vertical and with a speed

5 2 m/s. The projectile at the highest point of its trajectory splits into two equal parts. One part falls
vertically down to the ground, 0.5 s after the splitting. The other part, t seconds after the splitting, falls
to the ground at a distance x meters from the point O. The acceleration due to gravity g = 10 m/s2.

32. The value of t is _______. [IIT Advance 2021]

33. The value of x is _______.

Hints for Difficult Problems of Advance-II

Exercise -II
3. Use integration
5. Use momentum conservation and condition of complete circle.
7. Apply momentum conservation in horizontal direction. Energy released = change in KE.
8. Solve in the frame of C.M.
10. Apply momentum conservation in horizontal direction and work energy theorem

ANSWER KEY
EXERCISE–I
1. C 2. A 3. A 4. B 5. B 6. D 7. B

8. C 9. B 10. C 11. (a) C (b) B 12. B 13. D 14. B

15. C 16. D 17. C 18. A 19. A 20. C 21. B

22. C 23. B 24. C 25. A 26. D 27. (a) A (b) B

28. ABCD 29. (A) ® PQ ; (B) ® RS ; (C) ® PR ; (D) ® RS

EXERCISE–II
13 lx 5
1. 4 m/s, 24 J 2. v 3. (a) l(x) = l + , (b) L
2 0 L 9

5
4. g/9 downwards 5. (a) v0/3, (b) 3 5gR 6. cm, 153L/80u
17

7. (a) 360 m, (b) 10800 J 8. (a) v/2, v/2, 0; (b) 2mv2/9; (c) mv2/72; (d) x = m 6k v

2
mA 2gl 2m2A g
9. 2mv 3l 10. vB = mB 1 + m A mB ; T = 3mAg + mB

EXERCISE–III
1. C 2. D 3. B 4. B 5. A 6. C 7. C
8. D 9. A 10. A 11. A 12. C 13. A, D 14. B
–1
15. B 16. C 17. C 18. A 19. A 20. V = 4ms
21. A, C 22. D 23. 5 24. B 25. D 26. A 27. AD
28. C 29. C 30. 2.09 31. A, B 32. 0.45 - 0.55 sec 33. 7.30 - 7.70 meter
Vibrant Academy (I) Pvt. Ltd. "B-41" Road No.2, IPIA, Kota (Raj.) Ph. 06377791915 (www.vibrantacademy.com) [43]