PHYSICS - I B CENTRE OF MASS

m1z1 + m 2z 2 + ...... + m n z n
SYNOPSIS z cm =
m1 + m 2 + ...... + m n
1. Centre of mass of a body or a system of particles f) In Vector notation.
is the point at which the whole mass of the body
or system is supposed to be concentrated and If r1, r2 , r3 . . . are the position vectors of
moves as if the whole external force is applied
particles of masses m1, m2, m3 ...... then the
at that point.
position vector of their centre of mass is
2. The motion of centre of mass of a body represents m1 r 1 + m2 r 2 + ...... + mn r n
the motion of the whole body. r cm =
m1 + m2 + .... + mn
Two particles of masses m1 and m2 are separated g) Relative to centre of mass :
by a distance ‘d’. If x1 and x2 are the distances
m1 r1 + m 2 r2 + ....... + m r rn = 0 ,
of their centre of mass from m1 and m2, then
m1x1 = m2x2 where m 1 , m 2 .... m n have position
m2 d m1 d vectors r1, r2 .......rn relative to centre of mass i.e.
x1 = m + m and x2 = m + m
1 2 1 2 Algebraic sum of the moments of masses of a
b) The centre of mass of heavier and lighter mass system about its centre of mass is always zero.
system lies nearer to the heavier mass . h) In case of continuous distribution of mass to
c) Number of particles lying along the x-axis. locate the centre of mass we use.

Mò
x dm,Ycm = ò y dm, ZCM = ò z dm.
1 1 1
Particles of masses m1, m2, m3 ------- are at \ Xcm =
distances x1 x2 x3 ------ from the origin, the M M
distance of centre of mass from the origin. 4. Locating centre of mass can be done in following
ways.
m1x1 + m 2x 2 + m 3x 3 + ...... + m n x n
x cm = a) Method of symmetry
m1 + m 2 + m 3 + ....... + m n
Ex: I) For a circular hoop the centre of mass
d) Number of particles lying in a plane. lies at its centre.
Particles of masses m1, m2, m3, --------- are lying II) For a homogeneous sphere, centre of mass
in xy plane at positions (x1,y1), (x2,y2), (x3,y3) - lies at its geometric centre, because of its
------, then the position co-ordinates of their symmetry the sum of moments of mass about
centre of mass.
its centre is zero i.e., rdm = 0
m1x1 + m 2x 2 + ...... + m n x n
x cm = III) For an equilateral traingle the centre of mass
m1 + m 2 + ...... + m n is at its centroid.
m1y1 + m 2y 2 + ...... + m n y n IV) If two circular discs of radii r1 and r2 of same
y cm =
m1 + m 2 + ...... + m n material are kept in contact then the distance of
e)Particles distributed in space. centre of mass of system from centre of a disc
of radius r1 is given by
If (x1y1z1), (x2y2z2)...... are the position co-
ordinates of particles of masses m1, m2 ...... the r1 r2
r22 (r1 + r2 )
position co-ordinates of their centre of mass are x cm = 2 cm
m x + m 2x 2 + ...... + m n x n r1 + r22
x cm = 1 1
m1 + m 2 + ...... + m n
V) If two spheres of radii r1 and r2 of same
m1y1 + m 2y 2 + ...... + m n y n material are kept in contact, then the distance of
y cm =
m1 + m 2 + ...... + m n centre of mass of the system from centre of a

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y y
sphere of radius ‘r1’ is given by
r1 r2
r (r1 + r2 )
3 30 30
x cm =
3
2

r13 + r23
cm 20 20
10 10
1 2
0 x 0 x
10 20 30 10 20 30
VI) If two thin uniform rods of lengths L1 and
L2 of same material are joined to form ‘T’ shape C. Pappus theorems
as shown in the figure, then the distance of centre 1st law
of mass of the system from centre of mass of The volume ‘V’ generated as a result of the
first rod of length L1 is given by L revolution of a closed plane area about an axis
1
(such that every point moves perpendicular to
x cm
C the plane) equals to the area S of the palne times
L22 L2 the circumference (2 x c ) of the circle
x cm = C2
2(L1 + L 2 ) described by the centre of mass of the plane.i.e.,
V S 2 xc
VII) If two cylinders of lengths L1 and L2, radii Ex: Distance of centre of mass of a semi circular
r1 and r2 made up of same material are kept in
contact as shown in the figure, then distance of plate from its center is x c = 4r
3p
centre of mass of the system from the centre of 2nd law :
first cylinder is given by
The surface area S generated by revolution of a
2
æ L1 + L 2 ö plane curved line about an axis such that every
x cm =
r L2 çç ÷÷
(r L1 + r2 L2 )
2
2 2 ç
è 2 ø÷ point moves perpendicular to the line of the
1
curve is equal to the product of length of the
L1 L2 line and the distance through which the centre
C1
r1 x cm C2 r2 of mass moves.
VIII) If a wire of length ‘ ’ is bent in the form Ex : The distance of centre of mass of a wire
bent into the form of a half circle of radius R
a circular ring then the shift in center of mass is
2R
from its center is x c =
x= .
2p 5. Position of centre of mass of a body
IX) If a uniform rod of length L is bent at the a) depends on shape of the body.
mid point so that the two halves are inclined by b) depends on distribution of mass for a given
an angle q with each other then the shift in shape of the body.
æqö
center of mass, x = C os ççç ÷÷÷ .
L c) coincides with centre of gravity of the body
4 è2ø if the body is in uniform gravitational field.
X) Distance of centre of mass of a uniform cone 6. There may or may not be any mass at centre of
of height ‘h’ and base radius R, from the vertex mass. Ex uniform ring
3h Centre of mass may be within or outside the
on the line of symmetry is . material of body.
4
b) Method of decomposition a) For a circular disc the centre of mass is at
its geometric centre where there is mass.
Some of the surfaces can be divided into singular b) For a triangular plane lamina, the centre of
parts whose centre of mass can be found by the mass is the point of intersection of the medians
method of symmetry. of the triangle.

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7. If Dx1 , Dx2 are shift in position of 1st and 2nd 15. Shift in centre of mass when a small portion of
particle then shift in centre of mass along x - mass is removed from a uniform body is
axis is given by m removed d
x shift =
±m1Dx1 ± m2Dx2 (Mtotal - m removed )
Dxcm =
m1 + m2 16. Out of a uniform circular disc of radius R, if a
circular sheet of radius ‘r’ is removed then the
+ when shifted along + ve x – direction and
centre of mass of remaining part shifts by a
– when shifted along – ve x - direction
r 2d
simillary along y - axis distance where d is the distance of the
R2 r2
±m1Dy1 ± m2 Dy2 centre of the smaller part from the centre of the
Dycm =
m1 + m2 original disc.
If v1, v 2 , v 3 ....... v n are the velocities of (i) If circular portion is removed from the edge
particles of masses ,m1, m2, m3, ......, mn then r2
then, shift is maximum and xshift =
the velocity of their centre of mass. R +r

m 1 v 1 + m 2 v 2 + m 3 v 3 + ....... + m n v n (ii) If a circular portion of diameter R is removed

v cm =
m 1 + m 2 + m 3 + ....... + m n from a uniform circular plate of radius R, from
9. one edge then shift in the center of mass
M V cm = P 1 + P 2 + ..... + P n i.e total momentum of
R
the system is the product of mass of the whole xshift = .
system and the velocity of the centre of mass. 6
10. If two particles of masses m1 and m2 are moving (iii) If the centre of removed plate and the
with velocities v1 and v 2 at right angles to each original plate coincide then shift is zero.
other, then the magnitude of velocity of their 17. Out of a uniform solid sphere of radius R, if a
centre of mass is given by sphere of radius r is removed then the centre of
r 3d
m12 v12 + m 22 v22 mass of the remaining part shifts by
Vcm =
2
(R 3 r3)
(m1 + m 2 ) where d is the distance of the smaller sphere from
If a ,a ,a , ........ a n are the acccelerations of the centre of the original sphere.
1 2 3
particles of masses m1, m2, m3 .......mn then the (i) If spherical portion is removed from the edge
accelaration of their centre of mass is then, shift is maximum and
m 1 a 1 + m 2 a 2 + ....... + m n a n
a cm = r3
m 1 + m 2 + ....... + m n xshift = 2 2
m1 + m2 + m3 + .....+mn = M, total mass of the R + r + Rr
system, then m 1 a 1 +m 2 a 2 +.......m n a n = (ii) If a spherical portion of diameter R is
removed from a uniform sphere of radius R,
F1 + F2 + ...... + F n = F ext
from one edge then shift in the center of mass
M a cm = F ext R
12. Centre of mass can be accelerated only by a net xshift = .
external force. 14
13. Internal forces cannot accelerate the centre of 18. When a triangular portion is removed from one
mass or change the state of centre of mass. edge of a square plate of side ‘a’ as shown in the
14. In the absence of external forces, figure then shift in centre of mass is given by
a) the centre of mass of a system is at rest if the a
centre of mass is initially at rest.
a c1
xshift =
b) if the centre of mass of a system is moving a
c2
with constant velocity, it continues to move with 9
a 3
2
the same velocity.

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19. When a triangular plate is removed from one EXERCISE - I

edge of a rectangular plate of length a and
breadth b as shown in the figure then shift in 1. When no external force is acting on a system
a
centre of mass is given by xshift = of particles, the centre of mass of the system
a 9 1) remains at rest only
2) moves with constant velocity only
c1 c2 b
a
a
3 3) moves with constant velocity or will be at rest.
2
4) moves with variable velocity
20. When a shell in flight explodes,
2. Two particles, move towards each other from
a) The acceleration of centre of mass before and
rest under a mutual force of attraction. At
immediately after explosion is a cm = g the instant, when the speed of A is V, the speed
downward.
b) The centre of mass of all the fragments will of B is . The speed of centre of mass of the
continue to move along the same trajectory as
long as all the fragments are still in space. system is
c) If all the fragments reach the ground V 3 3
1) zero 2) 3) V 4) V
simultaneously, the centre of mass will complete 2 2 4
the original trajectory. 3. Three identical spheres each of radius R are
d) If some of the fragments reach the ground placed touching each other on a horizontal
earlier than the other fragments, the acceleration table. The centre of mass of the system is
located
of centre of mass changes .
1) at one of the centres of the spheres
21. When a person initially at rest in a boat at rest in 2) at the mid point joining the centres of any
still water, walks on a boat. Centre of mass of two spheres.
person + boat system is not displaced. 3) at the point of intersection of the medians of
a) If the man walks a distance L on the boat, the the triangle formed by the centres of the three
spheres.
boat is displaced in the opposite direction relative
4) at the mid point of a median of the triangle
to shore or water by a distance formed by the centres of the three spheres.
mL
x= 4. Two balls are thrown at the same time in
M+ m
(m = mass of man, M = mass of boat) vacuum. While they are in vacuum, the
accelaration of their centre of mass
b) distance walked by the man relative to shore 1) depends on masses of the balls
or water is (L-x) 2) depends on direction of motion of the balls
22. If two masses starting from rest move under 3) depends on speeds of the balls
mutual force of attraction towards each other, 4) is equal to accelaration due to gravity
then they meet at their centre of mass. 5. Centre of mass of a body (ATB)
a) in the above case, v cm = 0 and a cm = 0 1) always lies inside the body
b) If the two particles are m1 and m2 and their 2) always lies outside the body
3) always lies on the surface of the body
velocities are v1 and v 2 , then 4) may lie inside or outside the body
m1 v1 = –m2 v 2 6. Three identical masses are kept at the corners
c) If the two particles have accelerations of an equilateral triangle ABC. A moves
a1 and a 2 , then acm = 0 towards B with a velocity V, B moves towards
So, m1 a1 = –m2 a 2 C with velocity V, and C moves towards A
d) If s1 and s2 are the distances travelled before with same velocity V. Then the velocity of
centre of mass of the system of particles is
they meet
m1s1 = m2s2 1) V 2) zero 3) 3V 4)

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7. A shell moving in a parabolic path explodes. 14. A shell is thrown vertically up. The shell at
The centre of mass of the fragments move the highest point explodes into two equal
1) vertically down wards fragments. The centre of mass of the two
2) vertically upwards fragments
3) horizontally 1) goes further up and then falls
4) in the same parabolic path 2) falls down with an initial speed
3) falls down with zero inital velocity
8. A bomb at rest explodes. The centre of mass 4) comes to rest
of the system (ATB)
1) describes a parabola 15. Planets moving around sun and each planet
having satellites moving around them is an
2) vertically upwards
example of
3) horizontally 4) is at rest
1) rigid system
9. When an external force is applied at the 2) Non– rigid system
centre of mass of a system of particles, then 3) Both rigid and non rigid systems
it undergoes (ATB)
4) Neither rigid nor non – risid systems.
1) Only translatory motion
2) Only rotatory motion 16. Choose the wrong statement :
3) both translatory and rotatory motion 1) In the case of small bodies [Uniform
4) an oscillatory motion gravitional field] centre of mass and center
of gravity concide with each other.
10. A bomb moving in a parabolic path explodes
2) In non – uniform gravitational field centre
into two fragments of equal masses. The
of mass and centre of gravity will not
accelaration of the centre of mass of the
coincide with each other.
fragments when both are in air is equal to
1) g/2 2) 2g 3) g 4)zero 3) Centre of mass is independent of
aceeleration due to gravity, but center of
11. A cylinder is completely filled with water. If gravity depends on acelleration due to
1 gravity.
of the volume of water leaks out, its centre
4 4) Center of mass describes the stability of the
of mass,
1) moves up 2) moves down body where as centre of gravity describes
3) does not change the nature of motion of the body.
4) moves towards vertical surface 17. A thin uniform wire is bent in the form of a
12. Two bodies of masses m1 and m2 are at distances semi circle of radius R. Distance of its centre
x1 and x2 from their centre of mass. Then, the of mass from the geometric centre is
correct statement of the following is (ATB) 2R R
1) 2)
m1 x1 m1 x1
= = R R
1) 2) 3) 2 4) 2
m2 x2 m2 x2

m1 x2 m1 x2 18. Choose the correct statement (ATB)

= =
3) 4) 1) Centre of mass of two particles will be
m2 x1 m2 x1
nearer to lighter particle.
13. A uniform metre stick is placed vertically on 2) Centre of mass of the rigidbody depends on
a horizontal frictionless surface and released.
reference frame used.
As the stick is in motion, the centre of mass
moves 3) Centre of mass of the system of particles
1) vertically up 2) vertically down depends on the masses of the particles.
3) in a parabolic path 4) horizontally 4) Centre mass must lie with in the body.

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19. Distance of centre of mass of a thin uniform 25. A trolley filled with sand moves on a smooth
semi circular disc of radius R from its centre is horizontal surface with a velocity V0. A small
R 2R 4R 3R hole is made at the base of it from which sand
1) 2) 3) 3 4) 4 is leaking out vertically down at constant rate.
As the sand leaks out
20. Explosion is due to a) the velocity of the trolley remains constant
1) Internal forces 2) External forces b) the velocity of the trolley increases
3) Both 1 and 2 4) Neither 1 and 2 c) the velocity of trolley decreases
21. The path of the centre of mass of earth - moon d) the momentum of trolley + leaked out sand
system around the sun is is conserved
1) Circular path 2) Helical path 1) a & b are correct 2) c & d are correct
3) a & c are correct 4) b & d are correct
3) Parabolic path 4) Elliptical path
26. P is the centre of mass of a system of four
22. Choose the wrong statement (ATB)
point masses A, B, C and D, which are
1) In the process of explosion some changes
coplanar but not collinear
may occur in momentum of individual a) P may or may not coincide with one of the
fragments due to internal forces but the point masses
motion of the centre of mass is unaltered. b) P must lie within or on the edge of at least
2) Motion of centre of mass depends upon the one of the triangles formed by taking A, B, C
external force. and D three at a time
3) The location of centre of mass depends on c) P must lie on a line joining two of the points
the reference frame used locate it A, B, C, D
4) The position of centre of mass depends d) P lies out side the quadrangle ABCD
upon the shape of the body and the 1) a & b are correct 2) c & d are correct
distribution of mass . 3) a & c are correct 4) b & d are correct
23. A bomb travelling in a parabolic path under 27. Identify the correct statement. In the absence
the effect of gravity explodes in mid air. The of external force,
centre of mass of fragments will (1993 E) a) the kinetic energy associated with the
1) Move vertically upwards and then vertically motion of centre of mass is conserved
b) Centre of mass is not accelerated
downwards
c) The centre of mass always move in curved line
2) Move vertically upwards d) Centre of mass must be at rest
3) Move in an irregular path 1) a & b are correct 2) all are true
4) Move in the parabolic path the unexploded 3) all are false 4) a & c are correct
bomb would have travelled. 28. Mass of a ring is non-uniformly distributed
QUESTIONS ON MORE THAN ONE OPTION around its geometric centre. If R is radius of
the ring, then
24. a) Algebraic sum of moments of masses about a) Centre of mass does not coincide with
centre of mass is zero geometric centre
b) For small bodies centre of mass coincides b) Position of centre of mass from the
with centre of gravity geometric centre will be x (0 < x < R)
c) Position of centre of mass depends on co- c) Centre of mass will be nearer to the greater
ordinate system mass distribution
d) Position of centre of mass is independent d) Centre of mass may lie out side the
of mass distribution periphery
1) a and b are correct 1) only a and b are correct
2) b and c are correct 2) only b and c are correct
3) a, b and c are correct 3) a, b and c are correct
4) a, b, c and da are correct 4) a, b, c and d are correct

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29. A disc of radius 'r' is removed from the disc 33. Statement A : When no external force acts on
of radius 'R' then a body its centre of mass will be at rest or
a) The minimum shift in centre of mass is zero under uniform motion.
b) the maximum shift in centre of mass cannot Statement B : When a force acts on a body at
r2 its centre of mass then will have only translation
be greater than motion
( R + r)
c) Centre of mass must lie where mass exists 1) only A is correct
r2 2) only B is correct
d) the shift in centre of mass is 3) both A and B are correct
( R + r)
1) only a and b are correct 4) Both A and B are wrong
2) only a and c are correct 34. Statements :
3) only a, b and d are correct a) In translatory motion each point of the body
4) all are correct experiences the same displacement as any
30. In pure rolling motion of a ring other point as time goes on, so that the motion
a) it rotates about instantaneous point of of any particle represents the motion of the
contact of ring and ground whole body.
b) its centre of mass moves in translatory b) In rotatory motion, all the particles in the
motion only rigid body moves in concentric circles about
c) its centre of mass will have translatory as the axis of rotation.
well as rotatory motion 1) a is true, b is false
1) only a is correct 2) a and c are correct 2) a is false, b is true
3) a and b are correct 4) a, b and c are correct 3) both a and b are true
31. A tank filled with water is connected to another 4) both a and b are false
empty tank lying to the right side of the first 35. Statements :
tank. Untill the level of water in the tanks a) Centre of mass is that fixed point of a
become same system of particles or a rigid body with in the
a) centre of mass of water shifts down at boundaries of the system where the entire mass
constant rate
is concentrated.
b) centre of mass of water shifts down and
b) Center of gravity of a body is the point
rate of change of shift in centre of mass
through which the whole weight of the body acts.
decreases gradually and becomes zero
1) Both a and b are true
c) centre of mass shifts towards right as well
2) Both a and b are false
1) only a and c are correct
3) a is true, b is false
2) only b and c are correct
4) a is false, b is true
3) only a and b are correct
4) all are correct 36. Statements :
a) Position of C.m depends on acceleration due
32. Statement A : When an iron sphere on the
to gavity where as position of C.g independent
ground is heated its centre of mass rises
on aceeleration due to gravity.
slightly
b) Centre of gravity of a body is defined to
Statement B : Centre of mass of system of
know the amount of stability of the body and
particles depends on mass of the particles and
the center of mass of the body is defined to
their relative positions.
describe the nature of motion of the body.
1) only A is correct 1) a is true, b is false
2) only B is correct 2) a is false, b is true
3) A and B are correct 3) both a and b are ture
4) Both A and B are wrong 4) both a and b are false

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37. Statements : a) Ratio of the distances of the 42. Statements : a) Motion of centre of mass
centre of mass of a two particle system are in depends on external forces applied on it .
the inverse ratio of their masses : b) The motion of the centre of mass of the
b) The location of the centre of mass is body is called the translational motion of
independent of the reference frame used locate it the body.
1) Only a is true 1) Both a and b are true
2) Only b is true 2) Both a and b are false
3) Both a and b are true 3) a is true b is false
4) Both a and b are false 4) a is false b is true
38. The centre of mass of the system of particles 43. The moon moves around the earth in a
depends on circular orbit and the earth moves around
a) masses of the particles the sun in elliptical orbit then.
b) Relative position of the particles a) For the earth - moon system , the
1) Only a is true 2) Only b is true gravitational attractions between them are
3) Both a and b are true the internal forces
4) Both a and b are false b) The sun's attraction on the earth and
39. Statements : a) Allgebric sum of moments of moon are external forces acting on the
mass about centre of mass is equal to zero centre of mass of the system.
b) x - co ordinate of centre of mass of system 1) a is true b is false 2) a is false b is true
of particles in a plane is represented by 3) both a and b are true
1 4) Both a and b are false
x cm = mixi
M ASSERTION & REASON
C) x - co . ordinate of a rigidbody of continuous
mass distribution represented by These Questions consist of two statements each
printed as Assertion and Reason. While
1
x cm = x.dm answering these questions you are required to
M choose any one of the following four responses.
1) a and b are true 2) b and c are true 1) Both (A) and (R) are true and (R) is the
3) a and c are true 4) All a, b, c are true correct explanation of (A)
40. If no external force acts on the system 2) Both (A) and (R) are true and (R) is not the
a) acleleration of centre of mass is zero correct explanation of (A)
b) Velocity of centre of mass is zero 3) (A) is true but (R) is false
c) Centre of mass may be at rest (or) moving 4) (A) is false but (R) is true
with constant velocity 44. (A) : Position of centre of mass of a body
d) Velocity and momentum of centre of mass (w.r.t body frame)is independent of co-
remains constant ordinate system
1) a,b and c are true 2) a, c and d are true (R) : Internal forces do not effect the motion
3) b,c and d are true of centre of mass
4) a,b,c and d all are true 45. (A) : For the body shown in the figure , the
41. Statements : a) The motion of the centre of centre of mass lies outside the body
mass can be studied using kepler's laws of (R) : Centre of mass of a body may or may
motion. not lie within the body
b) The internal forces will not effect the motion
of the centre of mass
1) Both a and b are true 2) Both a and b are false
3) a is true b is false 4) a is false b is true

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46. (A) : Two particles ( starting from rest ) move less. The block starts from rest. The position
towards each other under a mutual force of of c.m of the system will change only in
attraction. The velocity of their c.m is zero vertical direction
(R) : Internal forces do not alter the state of
(R) : The system is free from external force
motion of c.m
along horizontal direction
47. (A) : When an external force is applied at the
centre of mass of a system of particles then it 53. (A) : There are some passengers in a
undergoes translatory motion only stationary compartment. A physical fight
(R) : The torque of the force acting at c.m, takes place between them because of different
about that c.m is zero opinions. The position of c.m of system will
48. (A) : When a person walks on a stationary not change
boat in still water, centre of mass of person (R) : The fight between passengers is internal
and boat system is not displaced force
(R) : Internal forces can not alter the position 54. (A) : Heavy boxes are loaded along with
of c.m. empty boxes on a cart. Heavy boxes should
49. (A) : A shell moving in a parabolic path be kept at the centre of gravity of the cart,
explodes in mid air. The centre of mass of the for more stability
fragments will follow the same parabolic path (R) : The vertical line passing through the
(R) : Explosion is due to internal forces, which centre of gravity should fall within the base
can not alter the state of motion of centre of of cart for stability.
mass 55. Assertion :(A) In the uniform gravitational
50. (A) : A vehicle is stopped due to friction field centre of mass and centre of gravity
between the wheels and the road coincide with each other. Where as in the case
of non – Unifrom gravitational field they will
(R) : Only external forces can cause the centre not coincide with each other.
of mass of system to accelerate or decelerate Reason : (R) Centre of mass is independent of
51. (A) : Stationary bomb explodes into two acceleration due to gravity but centre of
fragments. They move with different velocities gravity depends on acceleration due to
in opposite directions gravity.
(R) : Explosion does not violate the law of 56. Assertion : (A) Internal forces do not effect
conservation of linear momentum the path and position of centre of mass.
Reason : (R) According to newtons third law
52. (A) : A block of mass 'm' rests on a wedge of
of motion internal forces will occur in equal
mass M, which , in turn , is at rest on a
and opposite pairs . so they cancel each other.
horizontal table. All the surfaces are friction

MATCHING TYPE QUESTIONS

57. List–1 List–2
a) Centre of mass of large uniform ring e) mass is present
b) Centre of mass of non uniform rod f) Coincides with centre of gravity
c) Centre of mass of small funnel g) on the heavier side
d) Centre of mass of a uniform disc h) at Geometric centre
The correct match is
1) a – f; b–e; c – g; d – h 2) a – g; b– h; c–e; d – f
3) a – h; b – g; c – f; d – e 4) a – e; b – f; c – h; d – g

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58. List–1 List–2

a) Position of c.m of a triangle e)at one-forth of maximum height from the base
b) Position of c.m of a square f) at the point of intersection of medians
c) Position of c.m of a sphere g) at the point of intersection of diogonals
d) Position of c.m of a conc h) at the centre
The correct match is
1) a –f, b–g, c–h; d–e 2) a – g, b – h, c – e, d – f
3) a – g, b – h, c – f, d – e 4) a – g, b – f, c – e, d – h
59. List–1 List–2
a) Position of centre of mass e) is zero
b) The algebraic sum of moments of all
the masses about centre of mass f) in non. uniform gravitational field
c) Centre of mass and centre of gravity
coincide g) is independent of frame of reference
d) Centre of mass and centre of gravity do
not coincide h) in uniform gravitational field
The correct match is
1) a –e, b–g, c – f; d – h 2) a – g, b – e, c – f, d – h
3) a – g, b – e, c – h, d – f 4) a – h, b – e, c – f, d – g
60*. A square plate and a circular plate made up of same meterial are placed touching each other on
a horizontal table. If the side length of square plate is equal to diam eter of the circular plate then
the centre of mass of the system will be
1) inside the square plate 2) inside the circular plate
3) at the point of contact 4) outside the system

ANSWERS EXERCISE - II(A)

1) 3 2) 1 3) 3 4) 4 5) 4 (CLASS WORK)

6) 2 7) 4 8) 4 9) 1 10) 3 1. Centre of mass of two particles with masses

2 kg and 1 kg located at (1, 0, 1) and (2, 2,0)
11) 2 12) 3 13) 2 14) 3 15) 2
has the co-ordinates of
16) 4 17) 1 18) 3 19) 3 20) 1
21) 4 22) 3 23) 4 24) 1 25) 4 1) 2) 3) 4)

26) 1 27) 1 28) 3 29) 1 30) 3 2. Particles of masses 100 and 300 gram have
31) 2 32) 3 33) 3 34) 3 35) 1 position vector s (2ˆi + 5ˆj + 13kˆ ) and (-6ˆi + 4ˆj + 2kˆ ) .
36) 2 37) 3 38) 3 39) 4 40) 2 Position vector of their centre of mass is

41) 4 42) 1 43) 3 44) 2 45) 1 16 17 7 20 17 7

1) î + ĵ + k̂ 2) î + ĵ + k̂
46) 1 47) 2 48) 1 49) 1 50) 1 4 4 4 4 4 4

51) 1 52) 1 53) 1 54) 1 55) 1 16 17 19 16 13 19

3) î + ĵ + k̂ 4) î + ĵ + k̂
56) 1 57) 3 58) 1 59) 3 60) 1 4 4 4 4 4 4

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* 3. The distance between the centres of carbon and 9. Three particle each of 1kg mass are placed at
oxygen atoms in the carbon monoxide gas corners of a right angled triangle AOB, ‘O’
molecule is 1.13 x 10-10m. The distance of centre being the origin of coordinate system (OA and
of mass of the molecule relative to carbon atom is OB along the x-direction and +Ve
1) 0.48 x 10-10m 2) 0.64 x 10-10m y-direction). If OA = OB =1m, the position
3) 0.56 x 10-10m 4) 0.36 x 10-10m vector of centre of mass (in metres) is (2005 M)
4. The co-ordinates of centre of mass of particles
1)
iˆ + ˆj
2)
iˆ ˆj
3)
(
2 iˆ + ˆj ) 4) iˆ ˆj
of mass 10, 20 and 30 gm are (1, 1, 1) cm. The 3 3 3
position co-ordinates of mass 40 gm which 10. Four particles of masses 2,2,4,4 kg are
when added to the system, the position of arranged at the corners A,B,C,D of a square
combined centre of mass be at (0, 0, 0) are, ABCD of side 2m as shown in the figure. The
1) (3/2, 3/2, 3/2) 2) (-3/2, -3/2, -3/2) perpendicular distance of their centre of mass
3) (3/4, 3/4, 3/4) 4) (-3/4, -3/4, -3/4) from the side AB is,
5. Two particles of masses 4 kg and 6 kg are at
rest separated by 20 m. If they move towards
each other under mutual force of attraction,
the position of the point where they meet is
1) 12m from 4kg body 2) 12m from 6kg body
1) 1.33 m 2) 1 m 3) 1.5 m 4) 0.5 m
3) 8m from 4kg body 4) 10m from 4kg body
6. One end of a thin uniform rod of length L and 11. Two uniform rods A and B of lengths 5 m and 3
mass M1 is rivetted to the centre of a uniform m are placed end to end. If their linear densities
circular disc of radius 'r' and mass M2 so that are 3 kg/m and 2 kg/m, the position of their
both are coplanar. The centre of mass of the centre of mass from their interface is
combination from the centre of the disc is 1) 19/14 m on the side of heavier rod
(Assume that the point of attachment is at the 2) 8/7 m on the side of lighter rod
origin) (2003 M) 3) 2 m on the side of heavier rod
4) 2 m on the side of lighter rod.
L( M1 + M2 ) LM1
1) 2) 12. A uniform disc of radius R is put over another
2M1 2( M1 + M2 )
uniform disc of radius 2R of same thickness
2( M1 + M2 ) 2LM1 and density. The peripheres of the two discs
3) 4)
LM1 ( M1 + M2 ) touch eachother. The position of their centre
7. A ball of mass 3M rolls from rest on a smooth of mass is
flat table towards another ball of mass 2 M at
1) at from the centre of the bigger disc towards
rest with a constant velocity 5 m/s. When the
two balls are separated by a distance 6 m, the the centre of the smaller disc
position of centre of mass from the mass 3 M is
1) 3 m 2) 2.4m 3) 2.0 m 4) 1 m 2) at from the centre of the bigger towards

* 8. Three particles of masses 1 kg, 2 kg and 3 kg the centre of the smaller disc
are placed at the vertices A, B and C of an
equilateral triangle ABC. If A and B lie at (0,0) 3) at from the centre of the bigger disc
and (1,0) m, the co-ordinates of their centre towards the centre of the smaller disc
of mass are 4) none of the above
13. A system consists of two identical particles one
1) 2) particle is at rest and the other particle has an
acceleration 'a'. The centre of mass of the
system has an acceleration of (2002 M)
3) 4)
1) 2a 2) a 3) a/2 4) a/4

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14. Three identical spheres each of radius ‘R’ are 19. Three particles of masses 8 kg, 4 kg and 4 kg
placed touching each other so that their centres situated at (4,1), (-2,2) and (1,-3) are acted upon
A,B and C lie on a straight line. the position of by external forces 6 N, -6 N and 14 N. The
their centre of mass from A is accelaration of centre of mass of the system is
1) 0.625 ms-2 2) 6.25 ms-2
1) 2) 2 R 3) 4) 3) 2.2 ms-2 4) 22 ms-2
15. The velocities of three particles of masses 20g, 20. The mass of a uniform ladder of length 5 m is
30g and 50g are 10i, 10 j and 10k respectively. 20 kg. A person of mass 60 kg stands on the
ladder at a height of 2 m from the bottom. The
The velocity of the centre of mass of the three
position of centre of mass of the ladder and
particles is (2001 E) man from the bottom nearly is
1) 2 i + 3 j+ 5k 1) 1 m 2) 2.5 m 3) 3.5 m 4) 2.125 m
2) 10( i + j + k) 21. From a sphere of radius 1 m, a sphere of radius
0.5 m is removed from the edge. The shift in
3) 20 i + 30 j + 5 k the C.M. is
4) 20 i + 30 j + 50 k 1) 13/16 m 2) 16/13m
3) 14/13 m 4) 1/14 m
16. A thin uniform rod of length L is bent at its 22. A circular plate has a uniform thickness and
mid point as shown in the figure. The distance has a diameter 56 cm. A circular disc of
of centre of mass form the point ‘O’ is diameter 42 cm is removed from one edge of
the plate. The distance of centre of mass of the
L q L q remaining portion from the centre of the
1) cos 2) sin q circular plate is
2 2 4 2
1) 18 cm 2) 9 cm
L q L q 3) 27 cm 4) 4.5 cm
3) cos 4) sin
4 2 2 2 23. A man of 50kg is standing at one end on a boat
of length 25m and mass 200kg. If he starts
17. Particles of masses m, 2m, 3m,...... nm grams running and when he reaches the other end,
are placed on the same line at distances l, 2l, he has a velocity 2ms–1 with respect to the boat.
3l, .... nl cm from a fixed point. The distance The final velocity of the boat is (in ms–1)
of centre of mass of the particles from the fixed (2006 E)
point in centimetre is (2002 E) 2 2 8 8
(2n + 1)l l 1) 2) 3) 4)
5 3 5 3
1) 2)
3 n+ 1 24. Two particles having masses m and 2m are
n(n2 + 1)l 2l travelling along x-axis on a smooth surface with
3) 4)
2 n(n2 + 1) velocities u1 and u2, collide. If their velocities
18. Four particles, each of mass 1kg, are placed after collision are v1 and v2, then the ratio of
at the corners of a square OABC of side 1m. velocities of their centre of mass before and
‘O’ is at the origin of the coordinate system. after impact is
OA and OC are aligned along positive 1) 2 : 1 2) 2 : 3 3) 1 : 1 4) 1 : 2
x-axis and positive y-axis respectively. The
position vector of the centre of mass is 25. Two particles A and B initially at rest move
(in ‘m’) (2006 M) towards each other under a mutual force of

(iˆ + ˆj )
1 attraction. At the instant when the speed of A
1) iˆ + ˆj 2) is V and the speed of B is 2v, the speed of the
2
(iˆ ˆj )
centre of mass of the system is
(
3) iˆ ˆj ) 4)
1
2 1) 0 2) V 3) 1.5V 4) 3V

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26. A baloon of mass M is stationary in air. It has 33*. If the distance between the centres of the at-
a ladder on which a man of mass m is standing. oms of potassium and bromine in KBr molecule
If the man starts climbing up the ladder with a is 0.282 x 10-9 m, the centre of mass of this
velocity v relative to ladder, the velocity of two particle system from potassium is (mass
balloon is of bromine = 80 u and of potassium = 39 u).
m mv 1) 9.3 x 10-2 nm 2) 18.96 x 10-2 nm
1) v upwards 2) downwards -2
3) 27.9 x 10 nm 4) 3.3 x 10-2 nm
M (M + m)
*34. The co-ordinates of the centre of mass of the
mv Mv
3) upwards 4) downwards system as shown in figure are
(M + m) (M + m)
1) (0, 0)
27. A wooden log of mass 120kg is floating on still m
2) (1.33L, 0.167 L) Y=L
water perpendicular to the shore. A man of
mass 80kg is standing at the centre of mass of
3) (L, L) m m 3m
the log and he is at a distance of 30m from the x=0 x=L x=2L
shore. When he walks through a distance of 4) (0.33L, 1.67 L)
10m towards the shore and halts, now his
distance from the shore is *35. Two bodies of masses 10 kg and 20 kg are
located in x - y plane at (0, 1) and (1, 0). The
1) 20m 2) 24m 3) 28 4) 30m
position of their centre of mass is
28. A uniform wire of length L is bent in the form 1) (2/3, 1/3) 2) (1/3, 2/3)
of a circle. The shift in its centre of mass is 3) (2, 1) 4) (1/3, 4/3)
L 2L L L
1) 2) 3) 4) *36. A uniform disc of diameter R/2 is put over
2 3 another uniform disc of radius R of same
29. Masses 1 kg, 1.5 kg, 2 kg and M kg are situated thickness and density. The peripheries of the
at (2,1,1), (1,2,1), (2,–2,1) and two discs touch each other. The centre of mass
(–1,4,3). If their centre of mass is situated at of the system from centre of big disc is
(1,1,3/2), the value of M is 1) R/10 2) R/20 3) 9R/10 4) 11R/10
1) 1 kg 2) 2 kg 3) 1.5 kg 4) 3 kg *37. Particles of masses 1kg and 3kg are at
30. Three identical spheres each of radius ‘R’ are (2i + 5j + 13k) m and (–6i + 4j – 2k)m then
placed on a horizontal surface touching one instantaneous position of their centre of mass is
(-16i +17 j + 7 k ) m
another. If one of the spheres is removed, the 1
1)
shift in the centre of mass of the system is 4
2) (-8i + 17 j + 7k ) m
1
1) R/2 3 2) R/2 3) 3 R/2 4) R/ 3
4
3) (-6i + 17 j + 7 k ) m
31. Two bodies of masses 5kg and 3kg are moving 1
towards each other with 2ms –1 and 4ms –1 4
4) (-6i + 17 j + 5k ) m
respectively. Then velocity of centre of mass is 1
1) 0.25ms-1 towards 3kg 2) 0.5ms-1 towards 5kg 4
3) 0.25ms-1 towards 5kg 4) 0.5ms-1 towards 3kg 38. Two particles of same mass are projected
32. Two particles each of the same mass move due simultaneously with same speed 20 ms–1 from
north and due east respectively with the same the top of a tower of height 20m. One is
velocity ‘V’. The magnitude and direction of projected vertically upwards and other
the velocity of centre of mass is projected horizontally. The maximum height
V attained by centre of mass from the ground
1) NE 2) 2 V NE
2 will be (g = 10 ms–2)
V
3) 2 V SW 4) SW 1) 10 2m 2) 25 m 3) 25 2m 4) 5 m
2

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39. Six identical particles each of mass m are

arranged at the corners of a regular hexagon
EXERCISE - II(B)
of side length a. If the mass of one of the (HOME WORK)
particle is doubled, the shift in the centre of 1. The centre of mass of two particles with masses
mass is 2kg and 6kg located at (1, 1,1), (2, 2, 1)
respectively has the coordinates
6a
1) a 2) 7 7 7 4 4 2
7 1) , , ÷ 2) , , ÷
4 4 4 3 3 3
a a
3) 4) 7 7 7
7 3 3) , 4, 1÷ 4) , , 1÷
4 4 4
40. Three identical rods each of length L and mass
m are joined to form a U-shape as shown in 2. Two particles of masses 1kg and 3kg have
figure. The distance of centre of mass of the position vectors 2î + 3ĵ + 4k̂ and - 2î + 3ĵ 4k̂
combined system from the corner ‘O’ is respectively. The centre of mass has a position
vector (1999 E)
L L
1) 2) 1) î + 3ĵ 2k̂ 2) - î 3ĵ 2k̂
2 3
3) - î + 3ĵ + 2k̂ 4) - î + 3ĵ 2k̂
8L 13L 0 3. In a molecule of sodium chloride, the masses
3) 4)
6 6 of sodium and chlorine atoms are in the ratio
23 : 36. If the separation between the two
41. Four identical particles each of mass 1 kg are atoms is ‘d’, distance of centre of mass of the
arranged at the corners of a square of side molecule from sodium atom
length 2 2 m. If one of the particles is 1) 36d/59 2) 23d/59 3) 59d/23 4) 59d/36
removed, the shift in the centre of mass is * 4. The centre of mass of three particles of masses
1kg, 2kg, 3kg is at (2, 2, 2). The position of
8 4 2 fourth mass of 4kg to be placed in the system
1) m 2) m 3) m 4) 2m so that new centre of mass is at (0, 0, 0) is
3 3 3
(2005 E)
ANSWERS 1) (–3, –3, –3) 2) (–3, 3, –3)
3) (2, 3, –3) 4) (2, –2, 3)
1) 1 2) 3 3) 2 4) 2 5) 1 5. Two skaters A and B weighing 40 kg and 60
6) 2 7) 2 8) 3 9) 1 10) 1 kg respectively stand facing each other 5 m
apart. If they pull a light rope stretched
11) 1 12) 2 13) 3 14) 2 15) 1 between them, then they meet
1) 1.5 m from A 2) 2 m from A
16) 3 17) 1 18) 2 19) 1 20) 4
3) 2.5 m from A 4) 3.0 m from A
21) 4 22) 2 23) 1 24) 3 25) 1
6. A uniform bar 8 m long has a mass of 6 kg. A
26) 2 27) 2 28) 3 29) 3 30) 4 10 kg mass is tied to one end. The position of
centre of mass of the system is
31) 3 32) 1 33) 2 34) 2 35) 1
1) 1.5m from 10 kg mass
36) 1 37) 1 38) 2 39) 3 40) 4 2) 1.5 m from the centre of the rod
41) 3 3) 2.5 m from 10 kg mass
4) 4m from the free end of the rod

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7. Two persons of masses m and 2m are standing 12. Two discs of radii 2 cm and 1 cm and made up
on a horizontal smooth surface of ice. They of same material respectively are placed as
are initially seperated by a distance d. They shown below. The distance of centre of mass
of the system from O1 is
are pulling each other holding a rope. They
meet at a distance
1) 2d/3 from heavier person
2) d/3 from heavier person
3) 2d/4 from lighter person 1) 0.4 cm 2) 1.5 cm 3) 0.6 cm 4) 0.8 cm
4) d/3 from lighter person 13. Two particles of masses 2 m and 3 m are at a
distance ‘d’ apart. Under their mutual
8. Three particles of masses 1 kg, 2 kg and 2 kg
gravitational force, they start moving towards
are placed at the corners of an equilateral each other. The accelaration of their centre
triangle. The co-ordinates of A,B,C are (0,0) of mass when they are d/2 apart is
(2,0) and (x,y), the co-ordinates of their centre
of mass are 1) 2)

1) , 2) , 3) , 4) ,
3) 4) zero
9. Masses each 1 kg are placed at the verticies of an
isosceles triangle ABC in which AC=BC=5 cm and 14. Three identical particles each of same mass
are placed touching each other with their
AB = 8 cm. The distance of centre of mass of the
centres on a straight line. Their centres are at
system from the vertex C is A, B and C respectively. Then distance of
1) 2 cm 2) 1 cm 3) 1.5 cm 4) 3 cm centre of mass of the system from A is
10.Four particles P,Q,R,S of masses 1 kg, 2 kg, 3 kg AB + AC + BC AB + AC
1) 2)
and 4 kg are placed at the corners of a square of 3 3
side 1 m as shown below. the co-ordinates of the AB + BC AC + BC
3) 4)
centre of mass of the system are 3 3
15. Two objects of masses 200g and 500g possess
velocities 10 i ms–1and 3 i + 5 j ms–1 respectiv ely.
y.
–1
The velocity of their centre of mass in ms is
(2003 E)
5
1) 5 i 25 j 2) i 25 j
7
1) 0.5m, 0.7 m 2) 0.6 m, 0.7 m 25 5
3) 5 i + j 4) 25 i j
3) 0.9 m, 0.8m 4) 0.4m, 0.5 m 7 7
16. A uniform metre rod is bent into L shape with
11. Two uniform rods A and B of same diameter the bent arms at 90°to each other. The distance
having length 2m and 3m and having mass per of the centre of mass from the bent point is
unit 4kg m–1 and 6kgm–1 respectively are joined 1 1 1 1
end to end. The position of centre of mass of 1) m 2) m 3) m 4) m
4 2 2 2 2 8 2
combined rod from the free end of A is
17. Particles of masses 1gm, 2gm, 3gm and 4 gm
71 26
1) m 2) m are placed at x = 1 cm x = 2cm, x = 3 cm,
26 71
x = 4cm respectively. Then Xcm =
41 36
3) m 4) m 1) 1cm 2) 2cm 3) 3cm 4) 4cm
36 41

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18. Four particles, each of mass 1kg, are placed 24. Two particles of masses 1kg and 3kg move
at the corners of a square of side one meter in towards each other under their mutual force
the X – Y plane. If the point of intersection of of attraction. No other force acts on them.
the diagonals of the square is taken as the When the relative velocity of approach of the
origin, the coordinates of the centre of mass two particles is 2ms–1, the velocity of centre
are (2004 M)
of mass is 0.5ms –1 . When the velocity of
1) (1, 1) 2) (–1, 1) 3) (1, –1) 4) (0, 0)
approach becomes 3ms –1 , the velocity of
19. Three particles of masses 1kg, 2kg, 3kg are centre of mass is
acted upon by forces ( i + 2 j ) , ( 2 j + 3k ) and 1) 0.75 ms–1 2) 0.5 ms–1
( )
i k newton respectively. The magnitude 3) 2.5 ms–2 4) 0 ms–2
of acceleration of centre of mass to ms–2 is
25. Two identical particles move towards each
2
1) 3 2) 2 3 3) 3 4) 1 other with velocity 2V and V respectively. The
velocity of centre of mass is
20. The mass of a uniform ladder of length 6 m is
1) V 2) V/3 3) V/2 4) Zero
20 kg. A mass 60 kg is placed on the ladder at a
height of 2 m from the bottom. The position of 26. Two spheres of masses 2M and M are initially
centre of mass from the bottom is at rest at a distance R apart. Due to mutual
1) 1.5 m 2) 2.25 m 3) 2.25 m 4) 1 m force of attraction, they approach each other.
When they are at reparation R/2, the
21. A uniform sphere has radius R. A sphere of acceleration of the centre of mass of spheres
diameter R is cut from its edge as shown. Then would be
the distance of centre of mass of remaining 1) Zero 2) g ms–2 3) 3g ms–2 4) 12g ms–2
portion from the centre of mass of the original
sphere is 27. A boy of mass 36 kg moves from one end to
other end of a boat of mass 72 kg on still water.
1) R/7
If the length of the boat is 2.7m, the
2) R/14 R R
displacement of the boat is
3) 2R/7 1) 2.7 m 2) 1.35m 3) 5.4m 4) 0.9m
4) R/18 28. A uniform wire of length 6.28 cm is bent in the
22. A uniform circular disc has a radius of 20cm. form of a circle. The shift in its centre of mass is
A circular disc of radius 10 cm is cut 1) 1cm 2) 2cm 3) 4cm 4) 3.14 cm
concentrically from the original disc. The shift 29. A block of mass M is placed on the top of a
in centre of mass is bigger block of mass 10M as shown in fig. All
the surfaces are frictionless. The system is
1 7
1) cm 2) cm releated from rest. The distance moved by the
200 300 bigger block by the time the smaller block
1 reaches the ground is
3) cm 4) zero 1) 0.2m M
400
2) 0.4m
23. A man of mass 40kg in standing on a uniform 10M
3) 0.3m
plank of mass 60 kg lying on horizontal 4) 0.5m 2.2m
frictionless ice. The man walks from one end
30. Three identical spheres each of radius 3cm are
to the other end of the plank. The distance
placed on a horizontal surface touching one
walked by the man relative to ice is (given another. The distance of centre of mass of
length of plank = 5m) system from any centre of the sphere is
1) 2m 2) 3m 3) 5 m 4) 4m 1) 1/3cm 2) 3 cm 3) 2 3 cm 4)1/ 3 cm

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31. Two bodies of masses 4kg and 2kg are moving *37. Particals of masses 1kg and 3kg are at
towards each other with 3ms –1 and 2ms –1 (10i – 7j – 3k) ms–1 and (7i + 9j + 6k)ms–1 then
respectively. Then velocity of centre of mass is a instantaneous velociti of their centre of
1) 0.5ms-1 towards 4kg 2) 0.25ms-1 towards 2kg masses
3) 4/3 ms-1 towards 2kg 4) 4/3 ms-1 towards 4kg
(31i - 34 j +15k ) ms-1
1
32. Two particles of masses 3kg and 2kg move due 1)
4
north and due east respectively with the
(30i - 32 j +15k ) ms -1
velocities 2ms–1 and 3ms–1. The magnitude and 1
2)
direction of the velocity of centre of mass is 4
1) 1.2NE 2) 2 NE 3) 2 SW 4) 1.2 2 NE
(25i - 28 j +15k ) ms-1
1
* 33. The distance between the two nucleii of HCl 3)
4
is 1.27A. If the mass of chlorine atom is 37
(31i - 34 j + 20k ) ms -1
times that of Hydrogen, the centre of mass of 1
the system from the Hydrogen nucleus is 4)
4
1) 1.1A 2) 1.24A 3) 1.3A 4) 1.4A 38. Two particles of masses m1 and m2 (m1 > m2)
* 34.A body of mass 4 kg is attached to another are separated by a distance ‘d’. The shift in
body of mass 2 kg with a massless rod. If the 4 the centre of mass when the two particles are
kg mass is at (2i+5j)m and 2 kg mass is at interchanged.
(m1 + m 2 ) d (m1 - m 2 )d
(4i+2j)m, the centre of mass of the system is at (in m)
1
1) 8i + 12j 2) (8i + 12 j ) 1)
m1 - m 2
2)
m1 + m 2
2
1 1
3) (8i + 12 j ) 4) (8i + 12 j ) m1d m 2d
3 4 3)
m1 - m 2 4)
m1 - m 2
* 35.Two spheres of masses 4 kg and 8 kg aree
moving with velocities 2 ms–1 and 3 ms –1 away 39. Four identical particles each of mass m are
from each other along the same line. The arranged at the corners of a square of side
velocity of centre of mass is ‘a’. If mass of one of the particle is doubled,
8 -1 the shift in the centre of mass of the system is
1) ms towards second sphere
3 a a a a
1) 2) 3) 4)
8 -1 2 2 3 2 4 2 5 2
2) ms towards first sphere
3 40. If three rods of same mass are placed as shown
4 -1 in the figure, the co-ordinates of the centre of
3) ms towards second sphere mass of the system are
3
y
4 -1
4) ms towards first sphere
æ a a ÷ö æ a a ö÷
(0, a)
3 ç
1) ççç , ÷÷ 2) çç ÷
* 36. A disc of radius R is cut from a larger disc of è 2 2ø è 2 2 ø÷
,
radius 2R in such a way that the ed g e (a, 0) x
of the hole touches the edge of the disc. The
æ ö æ a a ÷ö
centre of mass of the remaining portion is at a a÷
3) ççç , ÷÷
ç
4) çç , ÷
1) R/2 from the centre of big disc away from the è 3 3ø è 3 3 ÷ø
centre of the hole
41. Six identicles particles each of mass 0.5 kg are
2) R/3 from the centre of big disc away from the
arranged at the corners of a regular hexagon
centre of the hole
of side length 0.5 m. If one of the particle is
3) 3R/2 from the centre of big disc away
removed, the shift in the centre of mass is
from the centre of the hole
4) 5R/2 from the centre of big disc away from 1 1 1 1
1) m 2) m 3) m 4) m
the centre of the hole 8 10 12 14

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4. Two steel spheres of radius r and 2r are made

ANSWERS to touch each other. The distance of their centre
of mass from their point of contact is
1) 4 2) 4 3) 1 4) 1 5) 4
8r
6) 1 7) 2 8) 2 9) 1 10) 2 1) at a distance in the bigger sphere
3
11) 1 12) 3 13) 4 14) 2 15) 3
2) at a distance r/3in the smaller sphere
16) 1 17) 3 18) 4 19) 3 20) 2
5r
21) 2 22) 4 23) 2 24) 2 25) 3 3) at a distance in the bigger sphere
3
26) 1 27) 4 28) 1 29) 1 30) 3 r
4) at a distance in the smaller sphere
31) 3 32) 4 33) 2 34) 1 35) 1 3
36) 2 37) 1 38) 2 39) 2 40) 3 5. Two particles of equal mass have velocities
41) 2 V1 = 4i and V2 = 4j ms–1. First particle has
EXERCISE - IIIA an acceleration a1 = ( 5i + 5j) ms –2 while the
acceleration of the other particle is zero. The
(CLASS WORK) centre of mass of the two particles moves in a
1. The C.M of a uniform card board cut in ‘T’ path of (2004 E)
shape as shown in figure is 1) Straight line 2) Parabola
3) Circle 4) Ellipse

* 6. Two blocks of masses 10 kg and 20 kg are placed

5 cm A B 2 cm on the x-axis. 10 kg mass is pushed on the
10 cm x-axis, by a distance of 2 cm towards the +ve x-
axis. The shift in the centre of mass is
2 cm 1) 2/3 cm 2) 2 cm
1) 4 cm from A towards B
2) 4 cm from B towards A 3) 1 cm 4) 6 cm
3) 3 cm from B towards A
7. A ladder AP of length 5m inclined to a vertical
4) 3 cm from A towards B
wall is slipping over a horizontal surface with
2. A metal rod of length 1 m is suspended from a
velocity of 2m/s, when A is at a distance 3m from
rigid support vertically at 00C. It is heated to
O, the velocity of CM at this moment is
200C. The linear coefficient of the material of
P
18 1) 1.5 ms–1 2) 2.5 ms–1
the rod is / K . The shift in the centre of 2m/s
106
mass is 3) 12.5ms–1 4) 1.25ms–1 A 3m O
18 9 2 3
1) 5
m 2) 5 m 3) m 4) m
10 10 10 5 10 5 8. A shell in flight explodes into n equal
3. On a large tray of mass M an ice cube of mass fragments, k of the fragments reach the ground
m, edge L is kept. If the ice melts completely, earlier than the other fragments. The
the centre of mass of the system come down by acceleration of their centre of mass
subsequently will be
1) 2) 1) g 2) (n-k)g

( n k )g ( n k)
3) 4) 3) 4) g
k n

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9. Four identical blocks of length L are arranged 14. From a uniform circular disc of radius 2cm
one over the other as shown. The maximum (its centre of mass is at O). A circular portion
distance of the uppermost block from the edge of radius 1cm is removed such that shift in
of the lowermost block is x such that no block centre of mass is maximum. The disc is now
tumbles. Then x is rotated about ‘O’ perpendicular to the plane
4L through then the magnitude of displacement
1)
3 1
3L of new centre of mass is cm, then is
2) 0 0
3 0
4 1)30 2) 45 3) 60 4) 1200
11L 15. Two balls of masses 5m and m have radii 2R
3)
12 and R. Their centre of masses are separated
by 12R. They move towards each other under
4) 2L their gravitational force. The distance moved
10. Four particles of masses m, 2m, 3m and 4m by the centre of smaller sphere when the
are at the verticies of a parallelogram in spheres touch each other is
x – y plane with one of the adjacent angle 600 1) 2. 5 R 2) 5R 3) 7.5 R 4) 10 R
and smaller side ‘a’ and larger side 2a. The
mass ‘m’ is at the origin and mass 4m x-axis. 16. A trolley of mass 360 kg is lying on a horizontal
The centre of mas of the system is frictionless surface. A man of mass 40 kg runs
with uniform speed on the trolley by a distance
3a a 2m 3m of 2m in 0.5 second. What is the velocity of the
1) ; ÷
2 2 a trolley relative to earth?
a 3a m 4m 1) 0.8 m/s 2) 0.4 m/s 3) 1.6m/s 4) 2.2m/s
; ÷ 2a
2) 17. A boat of mass 80 kg is floating on still water.
2 2
A dog of mass 20 kg on the boat is at a distance
3a 3a of 10m from the shore. The dog moves on the
3) 1.65a; 4 ÷÷ 4) 4
; 0.82a ÷
÷
boat by a distance of 2m towards the shore.
The distance of the dog from the shore is
11. Two blocks of masses 10 kg and 30 kg are
placed along a vertical line. The first block is 1) 11.6 m 2) 8.4 m 3) 9.6 m 4) 10.4 m
raised through a height of 7 cm. By what 18. A shell projected from a level ground has a
distance should the second mass be moved to range R, if it did not explode. At the highest
raise the centre of mass by 1 cm ? point, the shell explodes into two fragments
1) 1 cm upward 2) 1 cm downward having masses in the ratio 1:3; with each
3) 2 cm downward 4) 2 cm upward fragment moving horizontally immediately
12. A uniform square sheet has a side length of after the explosion. If the lighter fragment falls
2R. If one of the quadrants is removed, the at a distance R from the point of projection,
shift in the centre of mass behind the point of projection, the distance at
R R R which the other fragments falls from the point
3R
1) 2) 3) 4) of projection is
3 2 2 6 2 2 1) 2R 2) 5R/3 4) 4R/3 4) 2R/3
13. A uniform square sheet has a side length of
19. A shell is projected from a level ground with a
2R. A circular sheet of maximum possible area
is removed from one of the quadrants of the velocity of 20m/s at 45° to the horizontal. When
square sheet. The distance of centre of mass the shell is at the highest point it breaks into
of the remaining portion from the centre of two equal fragments. One of the fragments
the original sheet is whose initial velocity after the explosion is zero
R R falls vertically downward. At what distance
1) 2) from the point of projection does the other
2 [16 ] [16 ]
fragment fall? (g = 10 ms-2)
R R
3) 4) 1) 30 m 2) 60 m 3) 90 m 4) 40 m
[16 ] 16

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20. Two blocks of masses 2 kg and 1 kg respectively *24. A circular disc of diameter d and a square
are tied to the ends of a string which passes plate of side d are placed as shown in fig. The
over a light frictionless pulley. The masses are centre of mass of this combination from cen-
held at rest at the same horizontal level and tre of disc is (both the object are having same
then released. The distance transversed by mass per unit area)
centre of mass in 2 seconds is (g = 10 m/s2)
4d
1)
4+p
1) 1.42m
2) 2.22m 4d
2)
3+ p
d d
3) 3.12m
4) 3.33m 2 kg 1 kg 2d + 3p 3d + 7p
3) 4)
4+p 4+p
21. The linear density of a rod of length L varies
25. If a cube of side R/2 is removed from a
as = A + Bx where x is the distance from the uniform sphere of radius R as shown in the
left end. The distance of centre of mass from figure, the shift in centre of mass is
O is x 3R 2R

0 dx ( ) (
1) 2 8p -1 2) 3 8p -1
)
3A L + 2 BL2 2 A L + 2 BL2 R R
(
1) 3 2A + BL
) (
2) 3 2A + BL
) ( ) (
3) 3 8p -1 4) 4 8p -1
)
2 A L + 2 BL2 2 A L 2 BL2
(
3) 4 2A BL
) (
4) 4 2A BL
) ANSWERS
*22.The co-ordinates of centre of mass of letter F 1) 1 2) 1 3) 1 4) 3 5) 1
which cut from a uniform metal sheet are (take
orgin at bottom left corner) 6) 1 7) 4 8) 4 9) 3 10) 3

15 33 11) 2 12) 1 13) 1 14) 4 15) 3

1) , ÷ 16) 2 17) 2 18) 2 19) 2 20) 2
7 7
21) 1 22) 1 23) 2 24) 1 25) 2
15 33
2) , ÷
6 6
EXERCISE - III(B)
15 33 7 7 (HOME WORK)
3) , ÷ 4) , ÷
4 4 15 33
1. A rod consists of two parts made up of a
*23. A one meter long rod having a constant cross homogeneous material with square type cross
sectional area is made of four materials.The – section as shown. The position of C.M from
first 0.2m made of iron, the next 0.3 m of lead the end ‘O’ is
the following 0.2 m of aluminium and
remaining part is made of copper. If the
densities of iron lead, aluminium and copper 4 cm O 2 cm
10 cm
are approximately 8 x 103 kg/m3, 11 x 103 kg/
20 cm
m 3 , 3 x 10 3 kg/m 3 , and 9 x 10 3 kg/m 3
respectively, the centre fo mass of the rod from 35 65
the left end of iron rod is 1) cm 2) cm
3 3
1) 0.38 m 2) 0.48 m
55
3) 0.58 m 4) 0.68 m 3) cm 4) 15 cm
3
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2. A uniform rectangular plate is heated from 00 7. Two blocks of masses 10kg and 4kg are
to 1000C. Initial area of plate is 10 cm2. The connected by a spring of negligible mass and
linear coefficient of the material of the plate is placed on a frictionless horizontal surface. An
18 10 6 K . What is the shift of the centre of impulse gives a velocity of 14ms –1 to the
mass ? heavier block in the direction of the lighter
1) Zero 2) 1cm 3) 2cm 4) 3cm block. The velocity of the centre of mass is
3. Half of the uniform rectangular plate of length 1) 30ms–1 2) 20 ms–1
‘L’ is made up of material of density d1 and the 3) 10 ms –1
4) 5 ms–1
other half with density d2. The perpendicular
distance of centre of mass from AB is 8. A shell of mass m in flight explodes into three
equal fragments. One of the fragment reaches
the ground earlier than the other fragments.
The acceleration of the centre of mass of the
shell system just after first one reaches ground is
1) g 2) 2g/3 3) 3g/4 4) g/3
1) 2) 9. Seven homogneous blocks each of length L are
arranged as shown in th figure. Each block is
displaced with respect to the one in contact by
3) 4) L/10. The x-coordinate of the centre of mass
of the system relative to the origin O is
4. Two steel spheres of radius 2 cm and 5 cm are
made to touch each other. The distance of their
centre of mass from their point of contact is
1) at a distance 4 cm in the bigger sphere 11L 22L
1) 2)
2) at a distance 1.5 cm in the smaller sphere 12 25
3) at a distance 4.57 cm in the bigger sphere
0
4) at a distance 0.56 cm in the smaller sphere
22L 11L
5. Two particles of equal mass have velocities 3) 4)
35 35
V1 = 3i and V2 = 2j ms–1. First particle has
( )
an acceleration a1 = 5 i ms –2 while the
10. Masses 8, 2, 4, 2 kg are placed at the corners
A, B, C,D respectively of a square ABCD of
acceleration of the other particle is zero. The diagonal 80cm. The distance of centre of mass
centre of mass of the two particles moves in a from A is
path of 1) 20cm 2) 30cm 3) 40cm 4) 60cm
1) Straightline 2) Parabola 11. Consider a two-particle system with the
3) Circle 4) Ellipse particles having masses m1 and m2. If the first
particle is pushed towards the centre of mass
6. Two particles of masses 10kg and 30kg are through a distance d, by what distance should
lying on a straight line. The 10kg mass is the second particle be moved so as to keep the
shifted towards the 30kg mass by a distance centre of mass at the same position ?
of 2cm. By what distance should the 30kg mass (AIEEE 2004)
be shifted so that the position of their centre m1 m2
1) d = 2) d =
/ /
d d
of mass does not change m2 m1
1) 2/3 cm towards 10kg 2) 2/3 cm away from 10 kg m1 + m2 m1 m2
3) d = 4) d =
/ /
3) 3/2 cm towards 10 kg 4) 3/2 cm away from 10 kg d d
m1 m1

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12. A uniform square sheet has a side length of 12 18. A shell projected from a level ground has a
cm. If one of the quadrants is removed, the range R, if it did not explode. At the highest
shift in the centre of mass point, the shell explodes into two fragments
1) 1 cm 2) having masses in the ratio 1:2 ; with each
3 cm 3) 2 cm 4) 2 cm
fragment moving horizontally immediately
13. A uniform circular sheet has a radius r. A after the explosion. If the lighter fragment falls
square sheet whose diagonal length is equal to at a distance R/2 from the point of projection,
radius of the plate is removed. The maximum behind the point of projection, the distance at
distance of the centre of mass of the remaining which the other fragments falls from the point
plate from the centre of this original sheet is of projection is
r r
1) 2) 1) R/2 2) 2R/3 4) 4R/3 4) 7R/4
2( 2 1) ( 2)
r 19. A shell is projected from a level ground with a
2r
3) 4) velocity u at angle to the horizontal. When
3 2( 1) the shell is at the highest point it explodes into
14. The centre of mass coordinates of a block of two equal fragments. One of the fragments
shape shown in fig. is retraces its path to the point of projection. At
what distance from the point of projection does
Y the other fragment fall.
L L
2 u 2 Sin 2 u 2 Sin 2
2 2÷
1) ,
L 1) 2)
L g g
2
L 2 2
u Sin 2 2
2 u Sin 2
3) 4)
2g 3g
5 5
2) L, L÷ L
X
12 12 20. Two blocks of equal mass are tied with a light
string which passes over a massless pulley as
2 2 L L shown in figure. The magnitude of acceleration
3 3 ÷ 4 4÷
3) L, L 4) ,
of centre of mass of both the blocks is (neglect
15. Two balls of masses 2M and 6M have radii 2R friction everywhere)
and 3R. Their centre of masses are separated
by 10R. They move towards each other under 3 1
their gravitational force. The distance moved 1) g
by the centre of smaller sphere when the 4 2
spheres touch each other is
1) 5/4R 2) 7/2R 3) 10/3R 4) 15/4R
16. A man of mass 80 kg is riding on a small cart
2) ( 3 1g ) 600 300

of mass 40 kg which is rolling along a level

floor at a speed of 2 m/s. He is running on the g 3 1
cart so that his velocity relative to the cart is 3 3) 4) ÷g
2 2
m/s in the direction opposite to the motion of
cart. What is the speed of the centre of mass 21. The centre of mass of a non uniform rod of
of the system. Kx 2
1) 1.5 m/s 2) 1 m/s 3) 3 m/s 4) Zero length L whose mass per unit length = ,
L
17. A boat of mass 40 kg is floating on water. A where K is a constant and x is the distance from
boy of mass 10 kg on the boat moves by 8 m one end is
towards the shore. The distance by which the
boat moves away from the shore is 3L L K 3K
1) 2) 3) 4)
1) 4 m 2) 3 m 3) 2 m 4) 1.6 m 4 8 L L

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22. The co-ordinates of centre of mass of letter E

COMPREHENSION QUESTIONS
which is cut from a uniform metal sheet are
(Take origin at bottom left corner and width Comprehension - I (1 - 3) :
of letter 2 cm every where)
Two particles of masses 3 kg and 2 kg are at
1) (2cm, 4cm) (x 1, 0) and (x 2 , 0) where x 1 = 2t+3t 2 and
x2 = 5t–2t2, where x1, x2 are in metre and t is
2) (2.4cm, 5cm)
in seconds. Then
3) (3cm, 5cm)
1. Position of centre of mass at t = 2s is
4) 3.3cm, 5cm) 1) (0, 0) 2) (10.4, 0)
3) (52, 0) 4) (4.8, 0)
*23.A cylindrical rod of length 1.2 m is made of
2. Velocity of centre of mass at t = 2s is
three materials. The first part 0.5 m is made
1) zero 2) 4.8 ms–1
of iron, the second part 0.3 m is of copper and –1
the last part 0.4 m is of aluminium. If the 3) 7.2 ms 4) 9.6 ms–1
densities of iron, copper and aluminium are 8 3. Acceleration of centre of mass at t = 2s is
x 103 kg/m3, 9 x 103 kg/m3, respecitively the 1) zero 2) 2 ms–2
centre of mass of the rod from right end of the 3) 5.2 ms–2 4) 10 ms–2
aluminium rod is
1) 0.5 2) 0.67 m 3) 0.9 m 4) 1.2 m Comprehension - II (4 - 6):
Two blocks of masses 4 kg and 1 kg respectively
24. If a triangular sheet is removed from one edge are tied to the ends of a string which passes
of a uniform rectangular sheet as shown in the over a light frictionless pully. The masses are
figure, then the shift in the centre of mass of held at rest at the same horizontal level and
the system is then released. Then (g=10ms–2)
1) 3 cm
4. Acceleration of 4 kg block is
2) 6 cm 1) 6 ms–2 2) 5 ms–2
3) 6.6 ms –2 4) 4 ms–2
3) 9 cm
5. Acceleration of centre of mass of the blcoks is
4) 12 cm
1) 10 ms–2 2) 2.4 ms–2
25. If a square of side R/2 is removed from a uniform 3) 3.6 ms –2 4) 6 ms–2
circular disc of radius R as shown in the figure,
the shift in centre of mass is 6. Displacement of centre of mass of the blocks
in 2 sencods is
R R 1) 3.6 m 2) 7.2 m 3) 14.4 m 4) 12 m
1)
4p - 1 (
2) 2 4p -1
)
Comprehension - III (7 - 8) :
R R A Circular plate of diameter d is kept in contact
( ) (
3) 3 4p -1 4) 4 4p -1
) with a square plate of edge d as shown in figure.
The desnsity of the material and the thickness
ANSWERS are same every where.

1) 1 2) 1 3) 2 4) 3 5) 2 7. The centre of mass of the composite system

will be
6) 1 7) 3 8) 2 9) 3 10) 2
1) inside the circular plate
11) 1 12) 3 13) 1 14) 2 15) 4
2) inside the square plate
16) 4 17) 4 18) 4 19) 1 20) 1 3) at the point of contact
21) 1 22) 2 23) 2 24) 3 25) 4 4) outside the system

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8. Distance of centre of mass of the system from M and its length is L. Siva and Krishna can
the centre of the circular disc is exchange their positions in three different ways :
4d d d d Case–I : m1 moves towards B with Urel and m2
1) 4 2) 4 3) 1 4) 1 +
remains stationary until m1 reaches its position ;
Comprehension - IV (9 - 11): and then m2 starts moving and reaches the end A.
A particle of mass m moving horizontally with Case–II : m2 moves towards A with Urel and
v0 strikes a smooth wedge of mass M, as shown m1 remains stationary until m2 reaches its
in figure. After collision, the ball starts moving position, and then m 1 starts moving and
up the inclined face of the wedge and rises to a reaches the end B.
height h. Case–III : Both moves with Urel with respect
to trolley towards each other and reach then
v1 m v2 opposite ends.
M
12. Choose the wrong statement related to Case–I.
9. When the particle has risen to a height h on
1) As Siva moves, the trolley moves toward left
the wedge, then choose the correct alternative. and its velocity becomes maximum when he
1) The particle is stationary with respect to reaches the end B.
ground
2) Both are stationary with respect to the centre 2) When Siva reaches the end B, then distance
of mass. m1 L
3) The kinetic energy of the centre of mass moved by the trolley is
m1 + m2 + M
remains constant.
4) None of these 3) When Siva and Krishna has exchanged their
positions, then displacement of the centre of
10. Identify the correct statement related to the
mass of the system is zero.
situation when the particle starts moving
downward. 4) When the Siva and Krishna have exchanged
1) The centre of mass of the system remains their positions, then final velocity of the trolley is zero.
stationary. 13. Choose the correct statement related to Case–III.
2) Both the particle and the wedge remain
1) As both the men above moves simultaneously,
stationary with respect to centre of mass.
the velocity of the trolley at any instant is zero.
3) When the particle reaches the horizontal
2) Both men reach their opposite ends
surface its velocity relative to the wedge is v0.
simultaneously.
4) None of these
3) The distance travelled by both the men with
11. Choose the wrong statement related to the respect to ground is same.
wedge M. (v2 is final velocity of wedge) 4) All the above.
æ 4m2 ÷ö
= ç
ç ÷÷ gh 14. Choose the correct statement related to the
1) Its kinetic energy is f ç
K
è m + M ÷ø three cases.
æ 2m ö÷ 1) The displacement of the trolley cannot exceed L.
2) v2 = ççç ÷v
è m + M ø÷ 0 2) The displacement of the trolley is independent
3) Its gain in kinetic energy is of the velocity of each man.
æ 4mM ö÷æ 1 3) The displacement of the trolley in all three
ç
DK = çç ÷÷ç mv 2 ö÷÷
ç
cases is same.
çèç( m + M )ø÷÷çè 2
0÷
2
ø 4) All the above
4) Its velocity is less than the velocity of centre of mass. ANSWERS
Comprehension - V (12 - 14) : 1) 2 2) 3 3) 2 4) 1 5) 3
Two men Siva of mass m1 and Krishna of mass 6) 2 7) 2 8) 1 9) 2 10) 3
m2 are standing at the ends A and B of the
11) 4 12) 1 13) 2 14) 4
trolley, respectively. The mass of the trolley is
y
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