Physics Assignment Sheet

Topic  (Collision & Centre of Mass) SECTIONB

1. A uniform lamina ABCDE is made from a square ABDE and an
equilateral triangle BCD. Find the centre of mass of the lamina.
A B

O x
C

E a D

2. Find the distance of centre of mass of a uniform plate having

semicircular inner and outer boundaries of radii a and b from the
centre O.
a
b
O

3. Two particles A and B of mass 1 kg and 2 kg respectively are B

projected in the directions shown in figure with speed uA = 200 m/s
and uB = 50 m/s. Initially they were 90 m apart. Find the maximum uB
height attained by the centre of mass of the particles. Take g = 10 90m
uA
m/s2.
A

4. A gun (mass = M) fires a bullet (mass = m) with speed vr relative to barrel of the gun which is
inclined at an angle of 60º with horizontal. The gun is placed over a smooth horizontal surface.
Find the recoil speed of gun.
5. 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 figure. The height AC is 120 m. Canon balls of 1 kg
are fired from the car with an initial velocity 100 m/s at an
angle 30º with the horizontal. The first cannon ball hits the
stationary carriage after a time t0 and sticks to it. Determine
t0, the second cannon ball is fired. 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
cannon ball also hits and sticks to the carriage. What will be
the horizontal velocity of the carriage just after the second
impact? Take g = 10 m/s2.

6. A cylindrical solid of mass 10–2 kg and cross-sectional area 10–4 m2 is moving parallel to its axis (the
X-axis) with a uniform speed of 103 m/s in the positive direction. At t = 0, its front face passes the
lane x = 0. The region to the right of this plane is filled with stationary dust particles of uniform
density 10–3 kg/m3. When a dust particle collides with the face of the cylinder, it sticks to its surface.
Assuming that the dimensions of the cylinder remains practically unchanged, and that the dust sticks
only to the front face of the cylinder, find the x-coordinate of the front of the cylinder at t = 150 s. 6

7. A flat car of mass m0 starts moving to the right due to a

constant horizontal force F. Sand spills on the flat car
from a stationary hopper. The velocity of loading is
constant and equals to  kg/s. Find the time dependence
of the velocity and the acceleration of the flat car in the
process of loading. The friction is negligibly small.

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 F

8. A cart loaded with sand moves along a horizontal floor due to a constant force F coinciding in
direction with the cart’s velocity vector. In the process sand spills through a hole in the bottom with a
constant rate  kg/s. Find the acceleration and velocity of the cart at the moment t, at the initial
moment t = 0 the cart with loaded sand had the mass m0 and its velocity was equal to zero. Friction is
to be neglected.
9. (a) A rocket set for vertical firing weighs 50 kg and contains 450 kg of fuel. It can have a maximum
exhaust velocity of 2 km/s. What should be its minimum rate of fuel consumption. (i) to just lift off
launching pad.
(ii) to give it an acceleration of 20 m/s2.
(b) What will be the sped of the rocket when the rate of consumption of fuel is 10 kg/s after whole of
the fuel is consumed? Take g = 9.8 m/s2.

10. A uniform chain of mass m and length l hangs on a thread

and touches the surface of a table by its lower end. Find the
force exerted by the table on the chain when half of its length
has fallen on the table. The fallen part does not form heap.

ˆ ˆ ˆ
11. A ball of mass m, traveling with velocity 2i  3j receives an impulse 3mi . What is the velocity of
the ball immediately afterwards?
12. A particle of mass 2 kg is initially at rest. A F(N)
force starts acting on it in one direction whose
magnitude changes with time. The force time
graph is shown in figure. Find the velocity of 20
the particle at the end of 10 s.
10

t(s)
O 2 4 6 10

13. A string AB of length 2l is fixed at A to a point on a smooth A

horizontal table. A particle of mass m attached to B is
initially at a distance l from A as shown in figure. The
particle is projected horizontally with speed u at right angles l
to AB. Find the impulsive tension in the string when it
becomes taut and the velocity of the particle immediately
afterwards.
u
B

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14. Two particles A and B of equal mass m each are attached by a string of A
length 2l and initially placed over a smooth horizontal table in the
position shown in figure. Particle B is projected across the table with
speed u perpendicular to AB as shown in figure. Find the velocities of l
each particle after the string becomes taut and the magnitude of the
impulsive tension.

u
B

15. Two pendulum bobs of masses m and 2m collide elastically at the lowest point in their motion, when
the centres are at the same level as shown in the figure. If both the balls are released from height H
above the lowest point, to what heights do they rise for the first time after collision?
16. A ball is moving with velocity 2 m/s towards a heavy wall
moving towards the ball with speed 1 m/s as shown in figure.
Assuming collision to be elastic, find the velocity of ball 2m/s 1m/s
immediately after the collision.

17. After perfectly inelastic collision between two identical particles moving with same speed in different
directions, the speed of the particles becomes half the initial speed. Find the angle between the two
before collision.
18. A gun of mass M fires a shell of mass m and recoils horizontally. If the shell travels along the barrel
with speed v. Find the speed with which the barrel begins to recoil if :
(a) the barrel is horizontal
(b) The barrel is inclined at an angle 300 to the horizontal.
State in each case, the constant force required to bring the gun to rest in 2 second.
19. A cannon of mass m start sliding freely down a smooth included plane at an angle  to the horizontal.
After the canon covered the distance l, a shot was fired, the shell leaving the cannon in the horizontal
direction with a momentum P. As a consequence, the canon is stopped. Assuming the mass of the
shell to be negligible, as compared to that of the cannon, determine the duration of the shot.
ˆ ˆ
20. A smooth sphere of mass m is moving on a horizontal plane with a velocity 3i  j when it collides
with a vertical wall which is parallel to the vector ĵ . If the coefficient of restitution between the sphere
1
and the wall is 2 , find
(a) the velocity of the sphere after impact,
(b) the loss in kinetic energy caused by the impact
(c) the impulse ĵ that acts on the sphere.
ˆ ˆ
21. A sphere of mass m is moving with a velocity 4i  j when it hits a wall and rebounds with velocity
ˆi  3jˆ
. Find the impulse it receives. Find also the coefficient of restitution between the sphere and the
wall.
22. Two smooth spheres A and B, of equal radius but masses m and M, are free to move on a horizontal
table. A is projected with speed u towards B which is at rest. On impact, the line joining their centres
is inclined at an angle  to the velocity of A before impact. If e is the coefficient of restitution between
the spheres, find the speed with which B begins to move. If A’s path after impact is perpendicular to
eM  m
tan 2  
its path before impact, show that Mm .
23. Two smooth spheres, A and B, of equal radius, lie on a horizontal table. A is of mass m and B is of
ˆ ˆ ˆ ˆ
mass 3m. The spheres are projected towards each other with velocity vectors 5i  2 j and 2i  j
respectively and when they collide the line joining their centres is parallel to the vector î . If the
1
coefficient of restitution between A and B is 3 find their velocities after impact and the loss in kinetic
energy caused by the collision. Find also the magnitude of the impulses that act at the instant of
impact.

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24. A sphere A is of mass m and another sphere B of identical size but of mass 2m, move towards each
ˆ ˆ ˆ
other with velocity i  2 j and  i  3j respectively. They collide when their line of centres is parallel
i  ˆj e
1
to . If 2 , find the velocities of A and B after impact.
25. A simple pendulum is suspended from a peg on a vertical
wall. The pendulum is pulled away from the wall to a
horizontal position (see figure) and released. The ball hits the
2
wall, the coefficient of restitution being 5 . What is the
minimum number of collisions after which the amplitude of
oscillation becomes less than 60 degress?

26. Two blocks A and B of masses m and 2m respectively are B

C A
placed on a smooth floor. They are connected by a spring. A v0
m m 2m
third block C of mass m moves with a velocity v0 along the
line joining A and B and collides elastically with A, as
shown in figure. At a certain instant of time t0 after collision,
it is found that the instantaneous velocities of A and B are
the same. Further, at this instant the compression of the
spring is found to be x0. Determine (i) the common velocity
of A and B at time t0, and (ii) the spring constant.

27. A block A of mass 2m is placed on another block B of mass 4 m which in turn is placed on a fixed
table. The two blocks have the same length 4d and they are placed as shown in figure. The coefficient
of friction (both static and kinetic) between the block B and the table is . There is no friction
between the two blocks. A small object of mass m moving horizontally along a line passing through
the centre of mass of the block B and perpendicular to its face with a speed v collides elastically with
the block B at a height d above the table. (a) What is the minimum value of v(call it v0), required to
make the block A topple (b) If v = 2v0, find the distance (from the point P) at which the mass m falls
on the table after collision.

28. A bomb explodes in air when it has a horizontal speed of v. It breaks into two identical
pieces of equal mass. If one goes vertically up at a speed of 4v, find the velocity of other
immediately after the explosion.
29. Two identical buggies 1 and 2 with one man in each move without friction due to inertia
along the parallel rails towards each other. When the buggies get opposite each other, the
men exchange their places by jumping in the direction perpendicular to the motion direction.
As a consequence, buggy 1 stops and buggy 2 keeps moving in the same direction, with its
velocity becoming equal to v. Find the initial velocities of the buggies v1 and v2 if the mass of
each buggy (without a man) equal M and the mass of each man m.
30. A ball moving translationally collides elastically with another, stationary, ball of the same
mass. At the moment of the impact the angle between the straight line passing through the
centres of the balls and the direction of the initial motion of the striking ball is equal to
  45 . Assuming the balls to be smooth, find the fraction  of the kinetic energy of the
striking ball that turned into potential energy at the moment of the maximum deformation.

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31. Three identical discs A, B and C as shown in figure rests on a
smooth horizontal plane. The disc A is set in motion with v
A
velocity v after which it experiences an elastic collision
simultaneously with the discs B and C. The distance between
the centres of the latter discs prior to the collision is  times
greater than the diameter of each disc. Find the velocity of disc A after the collision. At what
value of  will the disc A recoil after the collision; stop, move on ?

32. The angular momentum of a particle relative to a certain point O varies with time as
→ → →2 → → → → →
j  a  bt , where a and b are constant vectors, with a  b . Find the force moment 
→ →
relative to the point O acting on the particle when the angle between the vectors
 and j
equal 45°.

33. A and B are two identical blocks of same mass 2 m and

A 2d
same physical dimensions. A is placed over the block B k
which is attached to one end of the spring of natural
B
length l and spring constant k. The other end of the spring d
is attached to a wall. The system is resting on a smooth
2d
horizontal with the spring in relaxed state. A small object
of mass m moving horizontally with speed v at height d above the horizontal surface hits the
block B along the line of their centre of mass in the perfectly elastic collision. There is no
friction between A and B. (a) Find the minimum value of V (say V0) such that block A will
topple over block B. (b) If V0 = V/2, find the period of oscillation of the block spring system
and amplitude. (c) What is the energy stored in the spring when the block B return to its
initial position as before collision ?

34. A small sphere of mass 10 g is attached to a point of a smooth vertical wall by a light string
of length 1 m. The sphere is pulled out in a vertical plane perpendicular to the wall so that
the string makes an angle 60° with the wall and is then released. It is found that after the first
rebound the string makes a maximum angle of 30° with the wall. Calculate the coefficient of
restitution and the loss of K.E. due to impact. If all the energy converted into heat, find the
heat produced by the impact.

35. Figure shows a uniform disc of radius R, from which a hole of radius y

R/2 has been cut out from left of the centre and is placed on right of
the centre of disc. Find the C.M. of the resulting disc.
(-R/2,0) (R/2,0) x
36. A solid circular cone of radius R is joined to a uniform solid
hemisphere of radius R. Both are made of same material. The centre O
R
of mass of the composite solid lies at the common base. Find the
height of cone.

R
O

37. The balloon, the light rope and the monkey shown in figure are at rest
in air. If the monkey reaches the top, by what distance does the
balloon descend ? Mass of balloon is M, mass of the monkey is m and
length of the rope ascended by the monkey is L.
38. A square hole is punched out from a circular lamina, the diagonal of
the square being a radius of the circle. Show that the centre of mass of
the remaining is at a distance R/(4  2) from the centre of the circle,
where R is the radius of the circular lamina.

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 R

a
O

39. A block of mass M with a semicircular track of radius R, rests on a horizontal frictionless
surface. A uniform cylinder of radius r and mass m is released from rest at the top point A as
shown in figure. The cylinder slips on a semicircular frictionless track. How far has the block
moved when the cylinder reaches the bottom (point B) of the track ? How fast is the block
moving when the cylinder reaches the bottom of the track ?
A
R

40. A ball of mass m = 1 kg is suspended from point ‘O’ of a toy cart of mass O A
M = 3 kg by an inextensible thread of length l = 1.5625 m. The ball is first I
I h
raised to point A and then the ball is released from rest. Point A is in the
same level as O and distance OA is such that the balls falls freely through
a height h = 1.25 m as shown in figure, then thread becomes taut.
Neglecting friction, calculate (a) velocity of cart just after the thread
becomes taut, and (b) loss of energy when thread becomes taut. (g = 10 ms2).

41. In the arrangement shown in figure pulleys are light and frictionless
and threads are flexible and inextensible. Mass of each of the
blocks A and B is m = 0.5 kg. Initially B is resting over a slab and A
is hanging. A shell of equal mass m = 0.5 kg and moving vertically A
upwards with velocity 0 = 12 ms1 strikes the block A and gets
h
embedded into at instant t = 0. Calculate (i) maximum height B
ascended by B when it is jerked into motion and time at that instant,
and (ii) time t when strikes the slab. Initial height of block A from the slab is h = 10 cm. (g =
10 ms2)
42. A ball of mass m = 1 kg is hung vertically by a thread of length 1 .5 0
m
l = 1.50 m. Upper end of the thread is attached to the ceiling of a m M
v0
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 0 = 6 ms1, collides
with the ball and gets stuck with it. As a result, thread starts to deflect towards right.
Calculate its maximum deflection with the vertical. (g = 10 ms2) m
43. A small ball of mass m is connected by an inextensible massless l
string of length l with an another ball of mass M = 4m. They are M
released with zero tension in the string from a height h as shown in h
figure. Find the time when the string becomes taut for the first time
after the mass M collides with the ground. Take all collisions to be elastic.
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