IITJEE Physics Centre of Mass Book
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CENTRE OF MASS
Contents Topic
Page No.
Theory
01 - 02
Exercise - 1
04 - 13
Exercise - 2
13 - 20
Exercise - 3
21 - 25
Exercise - 4
25
Answer Key
26 - 27
Syllabus Systems of particles ; Centre of mass and its motion ; Impulse ; Elastic and inelastic collisions.
Name :
Contact No.
CENTRE OF MASS CENTRE OF MASS MOMENTUM & COLLISION The action of force with respect to time is defined in terms of Impulse, that is, I=
Fdt = mv ñ mv f
i
=p
In the absence of a net external force, the momentum of a system is conserved.
dP
i.e.
dt = Fext = 0 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. A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved. In an inelastic collision, the total kinetic energy of the system changes. In a completely inelastic collision, the two bodies couple or stick togehter.
(i) (ii) (iii) 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. e=
rel. velocityof separation rel. velocityof 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.
1.
CENTRE OF MASS Discrete System : The position vector of the centre of mass is rc =
m1r1 m2 r2 ......... m n rn m1 m 2 .........mn
where r1, r2 ,..., rn are the position vectors of masses m1, m2, ...., mn respectively. The components of the position vector of centre of mass are defined as
mi x i xc 2.
=
M
mi yi ;
yc
=
M
mi zi ;
zc
=
M
Continuous system : The centre of mass of a continuous body is defined as
1 rc
M
r dm
In the component form
1 xc =
M
1
x dm
;
yc =
M
1
y dm
;
zc =
M
z dm
CENTRE OF MASS (Advanced) # 1
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 = (iii)
yc (iv)
4
A semi-circular ring
2R = ; xc = 0
A semi-circular disc
4R yc = ; xc = 0 3 (v)
A hemispherical shell
R yc = (vi)
2
; xc = 0
A solid hemisphere
3R yc 4. (i)
=
; xc = 0
8
Motion of the centre of mass : Velocity : The instantaneous velocity of the centre of mass is defined as
m i vi (ii)
(iii) (iv)
vc = M Acceleration : The acceleration of the centre of mass is defined as
mia i
ac = Mtotal momentum of a system of particles is Momentum : The p = Mvc Kinetic Energy : The kinetic energy of a system of particles consisits of two parts. K = Kc + Kí
1 where Kc = and
2
Kí =
2
Mvc , kinetic energy due to motion of c.m. relative to the fixed origin O,
1 2
2
, kinetic energy of the particles relative to the c.m.
mivi
Note that the term Kí may involve translational, rotational or vibrational energies relative to the centre of mass. 5.
Newonís 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.
Fext M acm
CENTRE OF MASS (Advanced) # 2
PART - I : OBJECTIVE QUESTIONS * Marked Questions are having more than one correct option.
SECTION (A) : CALCULATION OF CENTRE OF MASS A-1.
A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, where AB = AC = 5 cm. The third side BC, of length 6cm, is made from uniform wire of twice the density of the first. The distance of centre of mass from A is : (A)
11
34 11 cm
(B)
34
cm
11
(C)
34
cm
9
(D)
45
cm
A-2. All the particles of a system are situated at a distance r from the origin. The distance of the centre of mass of the system from the origin is (A) = r (B) r (C) > r (D) r A-3. A hemisphere and a solid cone have a common base. The centre of mass of the common structure coincides with the centre of the common base. If R is the radius of hemisphere and h is height of the cone, then (A)
h R
h
3
(B)
R
h
1 3
3
(D) h 1
(C) R
R
3
A-4. Five homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced with respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to the origin O shown.
33 L
(A) A-5.
11 L
(B)
25
25
22 L
(D)
(C) 25
33 L 50
ABC is a part of ring having radius R 2 and ADC is a part of disc having inner radius R 1 and outer R2. Part ABC and ADC have same mass. Then center of mass will be located, from the centre O.
(A)
(R2 ñ R1 )(2R1 R2 ) (above) 3(R1 R 2 )
(R2 ñ R1 )(2R1 R2 ) (B)
(below)
2R1 R2
(C) 2R1 R2 3
3(R1 R2 )
(above)
(D)
3
(below)
CENTRE OF MASS (Advanced) # 3
A-6.
From the uniform disc of radius R; an equilateral triangle of side 3 R is cut as shown in the figure. The new position of centre of mass is -
(A)
(0,0)
(B) (0, R)
3 R (C) 0, 2
(D) None of these
SECTION (B) : MOTION OF CENTRE OF MASS B-1. An object A is dropped from rest from the top of a 30 m high building and at the same moment another object B is projected vertically upwards with an initial speed of 15 m/s from the base of the building. Mass of the object A is 2 kg while mass of the object B is 4 kg. The maximum height above the ground level attained by the centre of mass of the A and B system is (take g = 10 m/s2) : (A) 15 m (B) 25 m (C) 30 m (D) 35 m B-2. Two particles having mass ratio n : 1 are interconnected by a light inextensible string that passes over a smooth pulley. If the system is released, then the acceleration of the centre of mass of the system is : n 1 g (B) n 1 2
(A) (n ñ
1)2
g
n 1 g (C) n 1 2
n 1 (D) n 1 g
B-3. Inside a smooth spherical shell of radius R a ball of the same mass is released from the shown position (fig.) Find the distance travelled by the shell on the horizontal floor when the ball comes to the just opposite position of itself with respect to its initial position in the shell.
3R
(A) B-4.
5
R (B)
4
(C)
3R 4
(D)
5R 4
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 wedge of mass M as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. Pulley is massless and frictionless. What is the displacement of the wedge when the block meets the pulley. (Man does not leave his position during the pull)
(A) 0.5 m
(B) 1 m
(C) Zero
(D)
23m
CENTRE OF MASS (Advanced) # 4
B-5. Two identical rods are joined at one of their ends by a pin. Joint is smooth and rods are free to rotate about the joint. Rods are released in vertical plane on a smooth surface as shown in the figure. The displacement of the joint from its initial position to the final position is (i.e. when the rods lie straight on the ground) :
(A) B-6.
L
(B)
4
(C)
5L 2
(D) none of these
Consider a thin stick of length L, standing on one of its ends on a frictionless surface. It is slightly pushed at the other end of the rod. Then, path of centre of mass of the rod is
2 (B) x2 + y2 = L 4
(A) x = 0
B-7.
17 L 4
(C) y = 0
(D)
x2 y2 1 2 L / 4 L 2
X and Y components of acceleration of C. M. are
(a
(A)
CM X
(a (C)
) m1 m2 g
) CM Y
m1 m2 m 2 2 m m g 1 2
m1m2g
(B)
(a
)
(m1 m 2 ) 2 m 2 (a ) g m m (D) CM Y 1 2 CM X
SECTION (C) : CONSERVATION OF LINEAR MOMENTUM C-1.
A block is kept at the top of a smooth wedge, which in turn is kept on a smooth horizontal surface. Then
(A) Horizontally the centre of mass will not shift (B) Centre of mass moves vertically (C) Centre of mass shift both direction horizontally as well as vertically (D) None of these
CENTRE OF MASS (Advanced) # 5
C-2.
If the resultant force on a system of particles is non-zero, then : (A) The linear momentum of the system must increase. (B) The velocity of the centre of mass of the system must change. (C) The distance of the centre of mass may remain constant from a fixed point. (D) kinetic energy of all particles must either increase simultaneuosly or decrease simultaneously.
C-3. A projectile is launched from the origin with speed v at an angle from the horizontal. At the highest point in the trajectory, the projectile breaks into two pieces, A and B, of masses m and 2m, respectively. Immediately after the breakup piece A is at rest relative to the ground. Neglect air resistance. Which of the following sentences most accurately describes what happens next?
(A) Piece B will hit the ground first, since it is more massive. (B) Both pieces have zero vertical velocity immediately after the breakup, and therefore they hit the ground at the same time. (C) Piece A will hit the ground first, because it will have a downward velocity immediately after the breakup. (D) There is no way of knowing which piece will hit the ground first, because not enough information is given about the breakup. C-4. A small bucket of mass M kg is attached to a long inextensible cord of length L m as shown in the figure. The bucket is released from rest when the cord is in a horizontal position. At its lowest position, the bucket scoops up m kg of water and swings up to a height h. The height h in meters is
(A)
M 2 L M m
M L M m
(B)
(C)
M m2 L M
M m L M
(D)
C-5.
A pendulum consists of a wooden bob of mass m and length
. A bullet of mass m 1 is fired towards v1 the pendulum with a velocity v 1. The bullet comes out of the bob with speed and the bob just 3 completes motion along a vertical circle. The velocity v1 is : m 3 m 3 m1 3 m 5g g 5g 5g (A) (B) (C) (D) 2 m m1 2 m1 2 m 1
SECTION (D) : SPRING - MASS SYSTEM .* When two blocks connected by a stretched spring (as shown) start moving from rest towards each other under mutual interaction. Then (pickup the correct alternative or alternatives)
(A) their velocities are equal and opposite. (B) their accelerations are equal and opposite. (C) the force acting on them are equal and opposite. (D) their momentum are equal and opposite CENTRE OF MASS (Advanced) # 6
D-2.
A block of mass m moving with a velocity v 0 collides with a stationary block of mass M at the back of which a spring stiffness k is attached, as shown in the figure. Choose the correct alternative(s)
(A) The velocity of the centre of mass is
v0 2 1 mM
v 20 .
(B) The initial kinetic energy of the system in the centre of mass frame is 4 M+ m mM 1 (C) The maximum compression in the spring is v 0 m + M k .
(D) When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero. D-3. Two blocks of masses m and M are moving with speeds v1 and v2 (v1 > v2) in the same direction on the frictionless surface respectively, M being ahead of m. An ideal spring of force constant k is attached to the backside of M (as shown). The maximum compression of the spring when the block collides is :
(A) v1
m k
(C) (v1 ñ v2)
(B) v2 mM (M m)K
M k
(D) None of above is correct
SECTION (E) : IMPULSE E-1. A gun which fires small balls of mass 20 gm is firing 20 balls per second on the smooth horizontal table surface ABCD. If the collision is perfectly elastic and balls are striking at the centre of table with a speed 5 m/sec at an angle of 608 with the vertical just before collision, then force exerted by one of the leg on ground is (assume total weight of the table is 0.2 kg and g = 10 m/s 2) :
(A) 0.5 N E-2.
(B) 1 N
(C) 0.25 N
(D) 0.75 N
A particle of mass m is made to move with a uniform speed v0 along the perimeter of a regular hexagon. The magnitude of impulse applied at each corner of the hexagon is
(A) 2mv0 sin
6
(B) mv0 sin
6
(C) mv0 sin
3
(D) 2mv0 sin
3
CENTRE OF MASS (Advanced) # 7
E-3.
A mass m connected to inextensible string of length l lie on a horizontal smooth ground. Other end of string is fixed. Mass m is imparted a velocity v such that string remains taut & motion occurs in horizontal plane. What is impulse provided by string during the time string turns through 90>.
(A) 2mv
(B) mv
(C)
2mv
(D)
m 2v2 2gl
SECTION (F) : COLLISION .* In an inelastic collision (external impulsive forces are absent) (A) The velocity of both the particles may be same after the collision (B) Kinetic energy of the system is not conserved (C) Linear momentum of the system is conserved. (D) Velocity of separation will be less than velocity of approach. F-2. Two particles of equal masses are moving with same speed collide perfectly inelastically. After the collision the combined mass moves with half of the speed of the individual masses. The angle between the initial momenta of individual particle is (A) 60 (B) 908 (C) 1208 (D) 458. F-3.
Which of the following statements is/are false ? (A) The magnitude of momentum of a heavy object is greater than that of a light object moving at the same speed (B) In a perfectly inelastic collision, all the initial kinetic energy of the colliding bodies is dissipated (C) The momentum of a system of colliding bodies may be conserved even though the total mechanical energy may not be (D) The velocity of the center of mass of a system is the systemís net momentum divided by its total mass
F-4.
A ball falls freely from a height h on to a smooth inclined plane forming an angle Assume the impact to be elastic. Then
with the horizontal.
(A) V0sin remains unchanged (where V 0 = velocity with which it strikes the plane) (B) Time of flight (T) for each collision remains unchanged. (C) Range on plane goes on increasing (D) Range on the plane goes on decreasing. F-5.
A ball of mass 1kg is dropped from a height of 3.2m on smooth inclined plane. The coefficient of restitution for
1 the collision is e =
. The ball's velocity become horizontal after the collision.
2
1 2
(A) The angle = tanñ1
(B) The speed of the ball after the collision =
3.2m
4 2 m/s
(C) The total loss in kinetic energy during the collision is 8J (D) The ball hits the inclined plane again while travelling vertically downward.
CENTRE OF MASS (Advanced) # 8
F-6.
A ball is dropped from a height h on the ground. If the coefficient of restitution is e, the height to which the ball goes up after it rebounds for the nth time is (A) h e2n
F-7.
(B) h en
(C)
e2 n h
(D)
h e2 n
Two smooth spheres A and B of equal radii but of masses 1 kg and 2 kg move with speeds 21 m/s and 4 m/s respectively in opposite directions and collide. The velocity of A is reduced to 1 m/s in the same direction. Then, which of the following statements is incorrect ? (A) The velocity of B becomes 6 m/s and its direction is
reversed
(B) The coefficient of restitution is 0.2 (C) The loss of kinetic energy of the system due to the collision is 200 J (D) The magnitude of impulse applied by the two spheres on each other is 10 Ns F-8.
Two blocks moving towards each other collides as shown in the figure. Find out the angle between the line of motion and the line of impact.
(A) 308 F-9.
(B) 608
(C) 908
(D) zero
Two identical balls A and B lie on a smooth horizontal surface, which gradually merges into a curve to a height 3.2 m. Ball A is given a velocity 10 m/sec to collide head on with ball B, which then takes up the curved path. The minimum coefficient of restitution 'e' for the collision between A and B, in order that B reaches the highest point C of curve. (g = 10 m/sec2)
(A)
1
3
(B)
2
(C)
5
1
(D)
4
3 4
F-10. A striker is shot from a square carrom board from a point A exactly at midpoint of one of the walls with a speed 2 m/sec at an angle of 45> with the x-axis as shown. The collisions of the striker with the walls of the fixed carrom are perfectly elastic. The coefficient of kinetic friction between the striker and board is 0.2. The coordinate of the striker when it stops (taking point O to be the origin) is :
1 (A)
2
2
,
1 2
1 (B) 0,
2 2
1 (C)
2 2
1 , 0
(D)
2
1 ,
2 2
CENTRE OF MASS (Advanced) # 9
SECTION (G) : VARIABLE MASS G-1.
An ice block is melting at a constant rate
dm dt
= µ. Its initial mass is m 0 and it is moving with velocity
v on a frictionless horizontal surface. The distance travelled by it till it melts completely is 2m0 v (A)
m0 v (B)
µ
m0 v (C)
µ
(D) can't be said
2µ
G-2. A balloon having mass ' m ' is filled with gas and is held in hands of a boy. Then suddenly it get released and gas starts coming out of it with a constant rate. The velocities of the ejected gases is also constant 2 m/s with respect to the balloon. Find out the velocity of the balloon when the mass of gas is reduced to half. (A) n 2
(B) 2 n 4
(C) 2 n 2
(D) none of these
G-3. A chain of length L and mass per unit length is pulled on a horizontal surface. One end of the chain is lifted vertically with constant velocity by a force P. (A) P as a function of height x of the end above the surface will be (gx + v2) (B) no energy will loss in this process 1 2 (C) work done by force will be gL 2 v 2 L 1 (D) loss in energy
gLv 2
2
PART - II : MISLLANEOUS QUESTIONS 1. COMPREHENSION COMPREHENSION # 1 Suppose a body of mass m 0 is placed on a smooth horizontal surface at rest. The mass of the body is decreasing exponentially with disintegration constant . Assuming that the mass is ejected backwards with a relative velocity u0 . 1.
The mass of body at an instant t is(A) m 0 eñt
2.
(B) m0 (1ñ eñt)
(B) u 0
(C) m0 u0 eñt
(D) Zero
The acceleration of the body is-
u0 0
4.
(D) None
The thrust force on the body is : (A) m 0 u0
3.
(C) m0 (1ñ t)
(A) u
(B)
u0
(C)
t
(D) Zero
2
The velocity - time graph is-
(A)
v
(B) t
v
(C) t
v
(D) t
v t
CENTRE OF MASS (Advanced) # 10
COMPREHENSION # 2
A ball of mass m = 1 kg is hung vertically by a thread of length = 1.50 metre. Upper end of the thread is attached to the ceiling of a trolley of mass M = 4 kg. Initially, trolley is stationary and it is free to move along horizontal rails without friction. A shell of mass m = 1 kg, moving horizontally with velocity v 0 = 6 m/s, collides with the ball and gets stuck with it. As a result, thread starts to deflect towards right. (g = 10 m/s2)
5.
Velocity of combined mass 2m just after collision is(A) 3 m/sec
6.
(B) 6 m/sec
(D) 1.5 m/sec
Velocity of the trolley, at the time of maximum deflection of the ball is (A) 3 m/sec
7.
(C) 1 m/sec
(B) 6 m/sec
(C) 1 m/sec
(D) 1.5 m/sec
Maximum inclination of thread with the vertical is (A) 30>
(B) 37>
(C) 45>
(D) 53>
COMPREHENSION # 3
Two blocks of masses m 1 and m 2 connected by an ideal spring of spring constant K are at rest on a smooth horizontal table. A constant horizontal force F acts on m1. During the motion maximum elongation of the spring is x0.
8.
Both blocks m 1 and m 2 move with same velocity when the elongation of the spring is (A)
m2 F 2K m m 1
9.
2
m2 F K m m 1
2
(C)
2m2 F K m m 1
2
(D)
4m2 F K m m 1
2
Both m 1 and m 2 move with same acceleration when elongation of spring is
m2 F
(A) 2K m m 1
10.
(B)
2
m2 F
(B) K m m 1
2
(C)
2m2 F K m m 1
2
(D)
4m2 F K m m 1
2
Select correct alternative(A) Velocity of center of mass is the same as that of m1 and m2 at the instant when elongation is x0 (B) Velocity of center of mass is same as that of m1 and m2 at the instant when elongation of spring is x0/2 (C) Velocity of center of mass can never be the as that of m1 and m2 at any instant (D) Velocity of center of mass can become zero at an instant during the motion of system
CENTRE OF MASS (Advanced) # 11
2. MATCH THE COLUMN 11.
Let
h0 be the initial height of ball with respect to the earth. The coefficient of restitution is e. Column I
Column II (P) e
(A) Total distance travelled by the ball
2n
h0
before coming to rest.
1 e2 (Q) h0
1 e2 1 e P (R) 1 e
(B) Height attained after n impacts (C) Average force exerted by ball
(S) mg
(D) Total momentum transferred to the earth
(T) None of these 12.
Assume that 2 bodies collide head on. The graph of their velocities with time are shown in column-I match them with appropriate situation in column-II Column-I Column-II
v (1)
(2)
m2
m1
(A)
(P)
m1 < m2 0 < e < 1
t
2nd body is large wall
v (1) m1 (B)
(Q)
(2) t
v (1)
(2) t
(C)
v (1) (D)
2
putty
(2) t
ball
(R)
(e = 0)
v1
v2
(S)
m1 = m2 e = 1 v1 > v2 m1
m2
(T)
m1 > m2
e=1
CENTRE OF MASS (Advanced) # 12
3.TRUE/FALSE 13.
(i) The magnitude of momentum of a heavy object is greater than that of a light object moving at the same speed. (ii) If net external force on a two body system is always zero, then direction of velocity of the centre of mass of given system may change. (iii) Internal forces can change, the momentum of a nonñrigid body.
4. FILL IN THE BLANKS Fill in the blanks 14.
(i) During the process of elastic collision kinetic energy first........... then.............. (ii) After completely inelastic collision of two blocks of different masses both the block will move with ............ component of .......... along line of collision.
PART - I : MIXED OBJECTIVE * Marked Questions are having more than one correct option. SINGLE CORRECT ANSWER TYPE 1.
The centre of mass of a system of particles is at (x , y , z ) where x 0, y 0. It is known that no particle 0
0
0
0
0
lies in the region y < 0 and x < 0 then the position of centre of mass can be (A) (0, 0, 4) (B) (0, ñ 4, 0) (C) (ñ 4, 0, 0) (D) ( ñ 4, ñ 4, 4,) 2.
A disc (of radius r cm) of uniform thickness and uniform density has a square hole with sides of r cm. One corner of the hole is located at the center of the disc and centre of the hole lies length = 2 on y-axis as shown. Then the y-coordinate of position of center of mass of disc with hole (in cm) is
(A)
3.
r 2( º)
(B)
r 4( º)
(C)
r 4( H)
(D)
3r 4( º)
A rigid system consists of two point masses, A and B of masses 1 kg and 2 kg respectively. At an instant the kinetic energy of A with respect to the centre of mass is 2 Joules and the velocity of centre of mass is 2 ms. The kinetic energy of the system at this instant is : (A) 9 J (B) 11 J (C) 13 J (D) none of these
CENTRE OF MASS (Advanced) # 13
4.
Two balls of same mass are released simultaneously from heights h & 2h from the ground level. The balls collides with the floor & sticks to it. Then the velocity-time graph of centre of mass of the two balls is best represented by :
(A)
5.
(D)
=g g may be less or more than g, depending on the masses of the rod and the insect.
A uniform rod OA of length resting on a smooth surface is slightly disturbed from its vertical position of unstable equilibrium. P is a point on the rod whose locus is a circle during the subsequent motion of the rod. Then the distance OP is equal to :
(A)
(B) 2
7.
(C)
A rod is allowed to fall freely under the influence of gravitational force. The rod remains vertical. An insect moves up the rod such that its velocity upwards relative to ground is constant. The acceleration of the rod is :
(A) (B) (C) (D) 6.
(B)
(D) there is no such point ++++++++
(C) 3
4
A small block of mass m is pushed towards a movable wedge of mass m and height h with initial velocity u. All surfaces are smooth. The minimum value of u for which the block will reach the top of the wedge
h u
(A)
2gh
(B)
2gh
(C)
m
1 2gh1
(D)
1 2gh1
CENTRE OF MASS (Advanced) # 14
8.
Which one of the following statements does not hold good when two balls of masses m 1 and m2 undergo elastic collision ? (A) when m1 < m2 and m2 at rest, there will be maximum transfer of momentum (B) when m1 > m2 and m2 at rest, after collision the ball of mass m2 moves with four times the velocity of m1 (C) when m1 = m2 and m2 at rest, there will be maximum transfer of K.E. (D) when collision is oblique and m2 at rest with m1 = m2, after collision the ball moves in opposite directions.
9.
A ball is bouncing down a set of stairs. The coefficient of restitution is e. The height of each step is d and the ball bounces one step at each bounce. After each bounce the ball rebounds to a height h above the next lower step. Neglect width of each step in comparison to h and assume the impacts to be effectively head on. Which of the following relation is correct ? (given that h>d) h
(A) 10.
h
=1 ñ
d
e2
(B)
= 1 ñ e
d
(C)
= d
1 e
h 2
(D)
1
= d
1 e
A ball of mass ëmí is released from the top of a smooth movable wedge of mass ëmí. When the ball collides with the floor, velocity of the wedge is ëví. Then the maximum height attained by the ball after an elastic collision with the floor is : (Neglect any edge at the lower end of the wedge).
2v2 (A) 11.
1
h
g
4v 2
v2 (B)
4g
(C)
g
v2 (D)
2g
A train of mass M is moving on a circular track of radius ' R ' with constant speed V. The length of the train is half of the perimeter of the track. The linear momentum of the train will be : 2MV
(A) 12.
0
(C) MVR
(D) MV
Four blocks of masses M1, M2, M3 and M4 are placed on a smooth horizontal surface along a straight line as shown. It is given that M1 >> M2 >> M3 >> M4. All the blocks are initially at rest. M1 is given initial velocity v0 towards right such that it will collide with M2. Consider all collisions to be perfectly elastic. The speed of M4 after all collision are over is
(A) v0 13.
(B)
(B) 4 v0
(C) 8 v0
(D) 16 v0
A spring is compressed between two blocks of masses m1 and m2 placed on a horizontal frictionless surface as shown in figure. When the blocks are released, they have initial velocity of 1v and v2 as shown in figure. The blocks travel distances x1 and x2 respectively before coming to rest. The ratio x 1 / x 2 is : m1 (A) m 2 m2 (B) m 1
(C)
m1 m2
(D)
m2 m1
CENTRE OF MASS (Advanced) # 15
14.
A system of two blocks A and B are connected by an inextensible massless strings as shown. The pulley is massless and frictionless. Initially the system is at rest when, a bullet of mass 'm' moving with a velocity 'u' as shown hits the block 'B' and gets embedded into it. The impulse imparted by tension force to the block of mass 3m is :
(A) 15.
5mu
(B)
4
4mu 5
(C)
2mu 5
(D)
3mu 5
The diagram shows the velocity - time graph for two masses R and S that collided elastically. Which of the following statements is true?
t (µs) I. R and S moved in the same direction after the collision. II. The velocities of R and S were equal at the mid time of the collision. 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 16.
Two equal masses are tied to the ends of a weighless inextensible thread passing over a weighless pulley. Initially the system is at rest and the masses are at the same level. A sharp horizontal impulse J is imparted to the right block as shown in figure. In subsequent motion
(A) the block A will come down relative to B (B) the block B will come down relative to A (C) the block will continue to be in the same horizontal plane. (D) the block A will remain at rest 17.
A hemisphere of mass 3m and radius R is free to slide with its base on a smooth horizontal table. A particle of mass m is placed on the top of the hemisphere. If particle is displaced with a negligible velocity, then find the angular velocity of the particle relative to the centre of the hemisphere at an angular displacement , when velocity of hemisphere is v -
4v (A) Rcos
(B)
3v Rcos
(C)
5v Rcos
(D)
2v Rcos
CENTRE OF MASS (Advanced) # 16
18.
Two particles A and B each of mass m are attached by a light inextensible string of length The whole system lies on a smooth horizontal table with B initially at a distance from A. The particle at end B is projected across the table with speed u perpendicular to AB. Velocity of ball A just after the jerk is -
(A) 19.
u 3 4
(B) u 3
(C)
u 3 2
(D)
u 2
AB is an L shaped obstacle fixed on a horizontal smooth table. A ball strikes it at A, gets deflected and restrikes it at B. If the velocity vector before collision is v and coefficient of restitution of each collision is 'e', then the velocity of ball after its second collision at B is (A) e 2 v
(B) e 2 v
(D) data insufficient
(C) ev 20.
In the figure shown a particle P strikes the inclined smooth plane horizontally and rebounds vertically. If the angle is 608, then the co-efficient of restitution is :
(A)
1
1 3
(B)
3
(C)
1 2
(D) 1
MULTIPLE CORRECT ANSWER(S) TYPE 21.
A block A of mass 1 kg is in contact with another block of same mass. A is attached to a spring of natural length 1 m and spring constant 100 N/m. The coefficient of friction for both of them is same ( = 0.2). The spring is initially compressed by 10 cm and released. What is a possible length of the spring when both blocks are in contact ?
(A) 90 cm 22.
(B) 95 cm
(C) 105 cm
(D) 103 cm
A series of n elastic balls whose masses are m, em, e2m, ....etc. are at rest separated by intervals with their centres on a straight line. Here, e is coefficient of restitution for the collision. The first is made to impinge directly on the second with velocity u. Then (A) The first (nñ1) balls will be moving with the same velocity (1ñe) u (B) The last one ball will move with velocity u 1 2 u (1 ñ e + en) (C) The kinetic energy of the system is 2 (D) None of these
CENTRE OF MASS (Advanced) # 17
23.
A ball is projected from a point in one of the two smooth parallel vertical walls against the other in a plane perpendicular to both after being reflected at each wall impinge again on the second at a point in the same horizontal plane as is started. The distance between two walls is a,b is the free range on a horizontal plane and e be the coefficient of restitution
(A) The total time taken in moving from O to C is
(B) The free range on the horizontal plane b =
a e2u
(e 2 + e + 1)
2uv g
(C) be2 = a (e2 + e + 1) (D) All above options are correct 24.
A sphere impinges directly on an identical sphere which is at rest. Then (A) First sphere comes to rest and second sphere move with same velocity if collision is perfectly elastic (B) After the impact both the sphere will combine and move together with common velocity if collision is perfectly inelastic (C) After the impact their velocities will be in ratio (1ñe) : (1 + e) if collision is partially plastic (D) Loss in kinetic energy =
1 2
(1 ñ e2) of original kinetic energy if collision is partially plastic
PART - II : SUBJECTIVE QUESTIONS 1.
A plate in the form of a semicircle of radius a has a mass per unit area of kr where k is a constant and r is the distance from the centre of the straight edge. By dividing the plate into semicircular rings, find the distance of the centre of mass of the plate from the centre of its straight edge.
2.
A small ring of mass m attached at an end of a light string the other end of which is tied to a small block B of mass 2 m. The ring is free to move on a fixed smooth horizontal rod. Find the velocity of the ring when the string becomes vertical.
3.
A small cube of mass m slides down a circular path of radius R cut into a large block of mass M. M rests on a table and both blocks move without friction. The blocks initially are at rest and m starts from the top of the path. Find the velocity v of the cube as it leaves the block.
CENTRE OF MASS (Advanced) # 18
A plate of mass M is moved with constant velocity v against dust particles moving with velocity u in opposite direction as shown. The density of the dust is and plate area is A. Find the force F required to keep the plate moving uniformly. ( in kg /m3)
5.
A particle moving on a smooth horizontal surface strikes a stationary wall. The angle of strike is equal 1 to the angle of rebound & is equal to 37> and the coefficient of restitution with wall is e = . Find the 5 X and fill value of X : friction coefficient between wall and the particle in the form 10
378 378
////////////////////////////////////
4.
6.
In the figure shown a small block B of mass m is released from the top of a smooth movable wedge A of the same mass m. The height of wedge A shown in figure is h = 100 cm. B ascends another movable smooth wedge C of the same mass. Neglecting friction any where the maximum height (in cm) attained by block B on wedge C is 20 + h . Find h
7.
Find the position of centre of mass of the uniform planner sheet shown in figure with respect to the origin (O)
CENTRE OF MASS (Advanced) # 19
8.
A disc of radius R is cut out from a larger uniform disc of radius 2R in such a way that the edge of the hole touches the edge of the disc. Locate the centre of mass of remaining part.
9.
A projectile is fired from a gun at an angle of 458 with the horizontal and with a speed of 20 m/s relative to ground. At the highest point in its flight the projectile explodes into two fragments of equal mass. One fragment, whose initial speed is zero falls vertically. How far from the gun does the other fragment land, assuming a horizontal ground ? Take g = 10 m/s2?
10.
A man of mass M hanging with a light rope which is connected with a balloon of mass m. The system is at rest in air. When man rises a distance h with respect to balloon Find. (a) The distance raised by man (b) The distance descended by balloon
11.
In a process a neutron which is initially at rest, decays into a proton, an electron and an antineutrino. The ejected electron has a momentum of p1 = 2.4 O 10ñ26 kg-m/s and the antineutrino p2= 7.0 O 10ñ27 kg-m/s. Find the recoil speed of the proton if the electron and the antineutrino are ejected (a) along the same direction. (b) in mutually perpendicular directions. (Mass of the proton m p= 1.67 O 10ñ27 kg.)
12.
A (trolley + child) of total mass 200 kg is moving with a uniform speed of 36 km/h on a frictionless track. The child of mass 20 kg starts running on the trolley from one end to the other (10 m away) with a speed of 10 m sñ1 relative to the trolley in the direction of the trolleyís motion and jumps out of the trolley with the same relative velocity. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run and just before jump?
13.
Two block of masses m1 and m 2connected with the help of a spring of spring constant k initially to natural length as shown. A sharp impulse is given to mass m2 so that it acquires a velocity v0 towards rigth. If the system is kept an smooth floor then find (a) the velocity of the centre of mass, (b) the maximum elongation that the spring will suffer
14.
During a heavy rain, hailstones of average size 1.0 cm in diameter fall with an average speed of 20 m/ s. Suppose 2000 hailstones strike every square meter of a 10 m O 10 m roof perpendicularly in one second and assume that the hailstones do not rebound. Calculate the average force exerted by the falling hailstones on the roof. Density of hailstones is 900 kg/m 3, take ( = 3.14)
15.
A ball of mass m moving at a speed v makes a head on collision with an identical ball at rest. The kinetic energy of the balls after the collision is 3/4 of the original kinetic energy. Calculate the coefficient of restitution.
CENTRE OF MASS (Advanced) # 20
PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS) * Marked Questions are having more than one correct option. 1.
A car P is moving with a uniform speed of 5 3 m/s towards a carriage of mass 9 kg at rest kept on the rails at a point B as shown in fig. The height AC is 120m. Cannon balls of 1 kg are shot from the car with an initial velocity of 100 m/s at an angle of 308 with the horizontal. The first cannon ball hits the stationary carriage after a time t 0and sticks to it. Determine t 0. Assume that the resistive force between the rails and the carriage is constant and ignore the vertical motion of the carriage throughout. If the second ball also hits and sticks to the carriage, what will be the horizontal velocity of the carriage just after the second impact ? [JEE- 2001, 10/100]
2.
Two particles of masses m1 and m2 in projectile motion have velocities u 1 and u 2 respectively at time t = 0. They collide at time t 0. Their velocities become v1 and v 2 at time 2t while still moving in air. The 0 value of [(m1v1 m2 v 2 ) (m1u1 m2 u 2 )] is [JEE (Scr) - 2001, 3/100] 1 (A) Zero (B) (m 1 + m 2)gt 0 (C) 2(m 1 + m2)gt0 (D) (m 1 + m 2 )gt 0 2
3.
Two blocks of masses 10kg and 4kg are connected by a spring of negligible mass and are placed on a frictionless horizontal surface. An impulse gives a speed of 14 msñ1 to the heavier block in the direction of the lighter block. Then, the velocity of the centre of mass is : [JEE 2002 Scr., 2/105] ñ1 ñ1 ñ1 (A) 30 ms (B) 20 ms (C) 10 ms (D) 5 msñ1
4.
A person at the origin O starts moving with a constant speed v 1along +y axis. At the same instant, a particle of mass m starts from point P with a uniform speed v 2along a circular path of radius R, as shown in figure. Find the momentum of the particle with respect to the person as a function of time t. [ JEE 2003, Mains, 2/60 ] y v1
v2 m
(0,0)
R
5.
(2R,0)
x
Two point masses m1 and m 2 are connected by a spring of natural length 0. The spring is compressed such that the two point masses touch each other and then they are fastened by a string. Then the system is moved with a velocity v 0 along positive x-axis. When the system reaches the origin the string breaks (t = 0). The position of the point mass m is given by x = v t ñ A(1 ñ cos t) where Aand are constants. Find the position of the second 1
1
0
block as a function of time. Also find the relation between A and 0.
[JEE-2003, 4/60]
CENTRE OF MASS (Advanced) # 21
6.
STATEMENT-1 In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision. because STATEMENT-2 In an elastic collision, the linear momentum of the system is conserved (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, 3/162]
7.
Two balls, having linear momenta p1 pài and p2 pài , undergo a collision in free space. There is no 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, 3/163] (A)
p'1 a1 ài b1 àj c 1 kà
p'1 c 1 kà
(B)
p'2 c 2 kà
p'2 a2 ià b 2 àj (C)
p'1 a1 ài b1 àj c 1 kà
p'1 a1 ài b1 àj
(D)
p'2 a2 ià b1 àj
p'2 a2 ià b2 jà c1 kà
Paragraph 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) Figure : [JEE-2008, 12/163] A
M
v 60>
B
30> 3m
8.
60 m/s
(B)
45 m/s
(C)
30 m/s
(D)
15 m/s
The speed of the block at point C, immediately before it leaves the second incline is (A)
10.
3 3m
The speed of the block at point B immediately after it strikes the second incline is (A)
9.
C
120 m/s
(B)
105 m/s
(C)
90 m/s
(D)
75 m/s
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 (A)
30 m/s
(B)
15 m/s
(C)
0
(D) ñ 15 m/s
CENTRE OF MASS (Advanced) # 22
11.
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 (ña, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, ña). The ycoordinate of the centre of mass of the ink in this drawing is : [JEE-2009, 3/160, ñ1]
a
(A) 12.
10
a
(B)
a
(C)
8
3
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 particles move with constant speeds. After making how many elastic collisions, other than that at A, these two particles will again reach the point A? [JEE-2009, 3/160, ñ1] v
(A) 4 13.
12
a
(D)
(B) 3
A
2v
(C) 2
(D) 1
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, 4/160,ñ1]
14*. A point mass of 1kg 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 msñ1. Which of the following statement(s) is (are) correct for the system of these two masses ? [JEE-2010, 3/163] (A) Total momentum of the system is 3 kg msñ1 (B) Momentum of 5 kg mass after collision is 4 kg msñ1 (C) Kinetic energy of the centre of mass is 0.75 J (D) Total kinetic energy of the system is 4 J 15. A ball of mass 0.2 kg rests on a vertical post of height 5 m. A bullet of mass 0.01 kg, traveling 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 20m 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
CENTRE OF MASS (Advanced) # 23
PART-II AIEEE (PREVIOUS YEARS PROBLEMS) * Marked Questions are having more than one correct option. 1.
Two identical particles move towards each other withvelocity 2vandv respectively. This velocityof centre of mass is ñ [AIEEE 2002, 4/300] (1) v (2) v/3 (3) v/2 (4) zero
2.
Consider the following two statements : [AIEEE 2003, 4/300] A. Linear momentum of a system of particles is zero B. Kinetic energy of a system of particles is zero, Then, (1) A does not imply B and B does not imply A (2) A implies B but B does not imply A (3) A does not imply B but B implies A (4) A implies B and B implies A
3.
Two particles A and B of equal masses suspended from two massless springs of spring constant k 1and k 2, respectively. If the maximum velocities, during oscillations are equal, the ratio of amplitudes of A and B is : [AIEEE 2003, 4/300] (1)
k1 / k 2
(2) k1 /k 2
(3)
k 2 / k1
(4) k1/k 2
4.
A rocket with a lift-off mass 3.5 O 104 kg is blasted upwards with an intial acceleration of 10 m/s2. Then the initial thrust of the blast is : [AIEEE 2003, 4/300] (1) 3.5 O 105 N (2) 7.0 O 105 N (3) 14.0 O 105 N (4) 1.75 O 105 N
5.
A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of mass 1
2
M and, a body C of mass M. The centre of mass of bodies B and C taken together shifts compared to 3 3 [AIEEE 2005, 4/300] that of body A towards: (1) depends on height of breaking (2) does not shift (3) shift towards body C (4) shift towards body B 6.
The block of mass M moving on the frictionless horizontal surface collides with the spring of spring constant k and compresses it by length L. The maximum momentum of the block after collision is : [AIEEE 2005, 4/300]
M
(1) 7.
Mk L
(2)
kL2 2M
(3) zero
(4)
ML2 k
A mass ëmí moves with a velocity ëví and collides in elastically with another identical mass. After collision the v in a direction perpendicular to the initial direction of motion. Find the 1st mass moves with velocity 3 speed of the 2nd mass after collision : [AIEEE 2005, 4/300] 2 (1) v
8.
9.
(2)
3v
(3)
v (4)
3 3 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 msñ1. The kinetic energy of the other mass is : [AIEEE 2006, 1.5/180] (1) 96 J (2) 144 J (3) 288 J (4) 192 J Consider a two particle system with particles having masses m and m . If2 the first particle is pushed 1 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, 3/180] (1) d
(2)
m2 d m1
m (3) m 1 d 1 m2
m1 (4) m d 2
CENTRE OF MASS (Advanced) # 24
10*.
A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs coincide. The centre of mass of the new disc is from the centre of the bigger disc. The value R of is : [AIEEE 2007, 3/120] (1) 1/3 (2) 1/2 (3) 1/6 (4) 1/4
11.
A body of mass m = 3.513 kg is moving along the x- axis with a speed of 5.00 msñ1 . The magnitude of its momentum is recorded as : [AIEEE 2008, 3/105] (1) 17.565 kg msñ1 (2) 17.56 kg msñ1 (3) 17.57 kg msñ1 (4) 17.6 kg msñ1
12.
A block of mass 0.50 kg is moving with a speed of 2.00 msñ1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is : [AIEEE 2008, 3/105] (1) 1.00 J (2) 0.67 J (3) 0.34 J (4) 0.16 J
13.
A thin rod of length 'L' is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/ x k n length) varies with x as , where n can be zero or any positive number. If the position xCM of the centre L of mass of the rod is plotted against 'n', which of the following graphs best approximates the dependence of xCM on n ? [AIEEE 2008, 3/105]
(1)
(2)
(3)
(4)
14.
Statement-1 : Two particles moving in the same direction do not lose all their energy in a completely inelastic collision. [AIEEE 2010, 4/144] Statement-2 : Principle of conservation of momentum holds true for all kinds of collisions. (1) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1. (2) Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1 (3) Statement-1 is false, Statement-2 is true. (4) Statement-1 is true, Statement-2 is false.
15.
A hoop of radius r and mass m rotating with an angular velocity 0is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip? [JEE Mains 2013] (1)
r0 4
r0
(2)
3
(3)
r0 2
(4) r0
NCERT QUESTIONS 1.
Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?
2.
In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 U (1 U = 10 -10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
3.
A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system ?
CENTRE OF MASS (Advanced) # 25
Exercise # 1 PART-I A-1.
(A)
A-2.
(B)
A-3.
(A)
A-4.
(D)
A-5.
(A)
A-6.
(B)
B-1.
(A)
B-2.
(C)
B-3.
(C)
B-4.
(A)
B-5.
(B)
B-6.
(A)
B-7.*
(BC)
C-1.
(AB)
C-2.
(BC)
C-3.
(B)
C-4.
(A)
C-5.
(A)
D-1.
(CD)
D-2.
(CD)
D-3.
(C)
E-1.
(B)
E-2.
(A)
E-3.
(C)
F-1.
(ABCD) F-2.
(C)
0F-3.
(BC)
F-4.
(ABC)
F-5.
(AB)
F-6.
(A)
F-7.
(D)
F-8.
(B)
F-9.
(B)
F-10.
(A)
G-1.
(B)
G-2.
(C)
G-3.
(ACD)
5.
(A)
06.
(C)
7.
(B)
PART-II 1.
(A)
2.
(C)
3.
(A)
4.
(A)
8.
(C)
9.
(B)
10.
(A)
11.
(A) Q, (B) P, (C) S, (D) R
12.
(A) S, (B) R, (C) Q, (D) P
13.
(i) T, (ii) F, (iii) F
14.
(i) Decrease, Increases, (ii) same, velocity
Exercise # 2 PART-I 1.
(A)
2.
(C)
3.
(A)
4.
(B)
5.
(C)
6.
(C)
7.
(C)
8.
(D)
9.
(C)
10.
(A)
11.
(B)
12.
(C)
13.
(B)
14.
(D)
15.
(D)
16.
(B)
17.
(A)
(A)
19.
(C)
20.
(A)
21.
(AB)
22.
(ABC) 23.
18.
4.
A(u + v)2
(ABCD) 24.
(ABCD)
PART-II
8g
3.
v =
2gR 1 m M
1.
3a/2
2.
V=
5.
5
6.
5
8.
At R/3 from the centre of the original isc away from the centre of the hole.
9.
60m
11.
(a)
13.
(a)
3 7.
(5a/6, 5a/6)
10.
mp
m2 v0 0 m m 1/ 2 (b) v (m 1 m 2 ) k
1
mh m M
(b)
Mh m M
2
p1 p 2 2
p1 p2 = 18.6 m/s (b) mp
m1 m 2
(a)
2
= 15.0 m/sec 12.
14.
9m/s, 9m 4 d3 3 8 vNA =1884 N
1 15.
e
=
2
CENTRE OF MASS (Advanced) # 26
Exercise # 3 PART-I
1.
100 3 m/ s t = 12 second; v = 11
2.
(C)
3.
(C)
4.
v 2 v 2 m(ñv sin R t ài + v cos R t àj ñ v àj) 2 2 1
5.
x
m1 A = 2 1 m
6.
(B)
7.
(A) an d (D)
8.
(B)
9.
(B)
(A)
(C)
13.
4
14.*
(AC)
15.
(D)
2
m1 2 m = v t + 2 A (1 ñ cos t), 10.
(C)
11.
0
12.
PART-II 1.
(3)
2.
(1)
3.
(3)
4.
(1)
5.
(2)
6.
(1)
7.
(3)
8.
(3)
9.
(4)
10.
(1)
11.
(4)
12.
(2)
13.
(4)
14.
(1)
15.
(3)
Exercise # 4 1.
The geometrical centre of each. No, the CM may lie outside the body,k as in case of a ring, a hollow sphere, a hollow cylinder, a hollow cube etc.
2.
Located on the line joining H and C1 nuclei at a distance of 1.24 U from the H end.
3.
The speed of the CM of the (trolley + child) system remains unchanged (equal to v) because no external force acts on the system. The forces involved in running on the trolley are internal to this system.
CENTRE OF MASS (Advanced) # 27
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