H C VERMA Solutions for Class 11 Physics Chapter 9 Centre of Mass, Linear Momentum, Collision

Question 1 Three particles of masses 1.0kg, 2.0kg and 3.0kg are placed at the corners A, B and C respectively of an equilateral triangle ABC of edge 1m. Locate the center of mass of the system.

Solution 1

Take A as origin (0,0) then C= (1,0)

if we take AC as x-axis. Similarly, if we take AB as x-axis,

Question 2 The structure of the water molecule in figure. Find the distance of the center of mass of the molecule from the center of the oxygen atom. Aak

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Solution 2

O Ao Take as origin. then

=0

Distance from O-atom=

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= m

m

Question 3 Seven homogeneous bricks, each of length L, are arranged in shown in figure. Each brick is displaced with respect to the one in contact by L/10. Find the x-coordinate of the center of mass relative to the origin shown.

Solution 3

take O as (0,0)

brick A and E is from

brick B and D is from

brick C is from ,F are from (0 to L)

For and F bricks, C.O.M=

For B and D, C.O.M=

(as they are displaced by ) Similarly,

For A and E C.O.M=

For C, C.O.M=

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of all bricks

Question 4 A uniform disc of radius R is put over another uniform disc of radius 2R of the same thickness and density. The peripheries of the two discs touch each other. Locate the center of mass of the system.

Solution 4

C.O.M of 2R disc =(2R,0) C.O.M of R disc=(R,0)

C.O.M of system= }

{ }

C.O.M of system=

= Distance of C.O.M (system) from C.O.M of 2R disc

=

Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 160 Question 5 A disc of radius R is cut out from a larger disc of radius 2R in such a way that the edge of the hole touches the edge of the disc. Locate the center of mass of the residual disc.

Solution 5 Aakas

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C.O.M of 2R disc =(2R,0) C.O.M of R disc=(R,0)

C.O.M of shaded area=

=

From C.O.M of 2R disc, it is distance away.

Question 6 A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the center of mass of the combination, assuming same mass per unit area for the two plates.

Solution 6

Area density of square= (Mass/unit area)

Mass/unit area of circle=

C.O.M of square = ( )

C.O.M of circle = ( )

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C.O.M of the whole system=

=

= From C.O.M of disc it is

= distance away.

Question 7 Calculate the velocity of the center of mass of the system of particles shown in figure.

Solution 7 Velocity coordinates of 1Kg For another 1Kg For 1.2Kg For 1.5Kg For 0.5Kg

COM= (0.13487, -0.13915)

Question 8

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Two blocks of masses 10Kg and 20Kg are placed on the X-axis. The first mass is moved on the axis by a distance of 2cm. By what distance should the second mass be moved to keep the position of the center of mass unchanged?

Solution 8

Initially, let 10Kg be cm away 20Kg be cm away

Let 20Kg be moved by distance of 2cm.

cm

20Kg should move 1cm towards left.

Question 9 Two blocks of masses 10Kg and 30Kg are placed along a vertical line. The first block is raised through a height of 7cm. By what distance should the second mass be moved to raise the center of mass by 1cm?

Solution 9

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cm should be moved 1cm downward.

Question 10 Consider a gravity-free hall in which a tray of mass M, carrying a cubical block of ice mass m and edge L, is at rest in the middle. If the ice melts, by what distance does the center of mass of "the tray plus the ice" system descend?

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Solution 10

As ice melts, COM would no shift, as there is no external force (not even gravity).

Question 11 Find the center of mass of a uniform plate having semicircular inner and outer boundaries of radii and .

Solution 11

C.O.M of shaded region=

{ =density of material} {l=thickness of disks}

C.O.M of shaded area

=

=

=

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= above center

Question 12 Mr. Verma (50Kg) and Mr. Mathur (60Kg) are sitting at the two extremes of a 4m long boat (40Kg) standing still in water. To discuss a mechanic problem, they come to the middle of the boat. Neglecting friction with water, how far does the boat move on the water during the process?

Solution 12

V-Verma M-Mathur B-Boat

m When they come to center

m

Shift in

m So, the boat would shift 0.13m so that COM of system remains same.

Question 14 The balloon, the light rope and the monkey shown in figure are at rest in air. If the monkey reaches the top of the rope, by what distance does the balloon descend? Mass of the balloon=M, mass of the monkey=m and length of the rope ascended by the monkey=L.

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Solution 14

Let balloon be at origin.

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Both monkey and balloon are at origin.

Shift=

=

So, the balloon descends by distance.

Question 15 Find the ratio of the linear momenta of two particles of masses 1.0Kg and 4.0Kg if their kinetic energies are equal.

Solution 15

Question 16 A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of m/s. Calculate the recoil speed of the residual nucleus thorium-234. Assume that the mass of a nucleus is proportional to the mass number.

Solution 16 Aak

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m/s

Question 17 A man of mass 50Kg starts moving on the earth and acquires a speed of 1.8m/s. With what speed does the earth recoil? Mass of earth= Kg.

Solution 17 Conservation of lin. Momentum (C.O.L.M)

M-Man E-Earth

m/s

Question 18 A neutron initially at rest decays into a proton, an electron and an antineutrino. The ejected electron has a momentum of Kg-m/s and the antineutrino Kg-m/s. Find the recoil speed of the proton (a) if the electron and the antineutrino are ejected along the same direction and (b) if they are ejected along perpendicular directions. Mass of the proton= Kg.

Solution 18 n

Kg-m/s Kg-m/s

(a) If and are along same direction.

Resultant velocity, in same direction.

C.O.L.M

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m/s

(b) If , or

m/s

Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 161 Question 19 A man of mass M having a bag of mass m slips from a roof of a tall building of height H and starts falling vertically. When at a height h from the ground, he notices that the ground below him is pretty hard, but there is a pond horizontal from the distance x from the line of fall. In order to save himself he throws the bag horizontally in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in water. If the man just succeeds to avoid hard ground, where will the bag land?

Solution 19 When he throws a bag to left, momentum is conserved.

velocity of bag speed of man

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Let pond start from ( )

Time to reach height h=

Time to reach ground,

Time taken by man=

= COM of the system will be at (0,0) after he reaches ground (let bag reach from origin) ( )

From law of kinematics

Question 20 A ball of mass 50g moving at a speed of 2.0m/s strikes a plane surface at an angle of incidence 45°. The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate (a) the magnitude of the change in momentum of the ball. (b) thechange in the magnitude of the momentum of the ball.

Solution 20 Aak

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(a) change in momentum

kg-m/s (b) change in momentum magnitude

Question 21 Light in certain cases may be considered as a stream of particles called photons. Each photon has a linear momentum h/ where h is the Planck's constant and is the wavelength of the light. A beam of light of wavelength is incident on a plane mirror at an angle of incidence . Calculate the change in the linear momentum of a photon as the beam is reflected by the mirror.

Solution 21 Aak

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Question 22 A block at rest explodes into three equal parts. Two parts starts moving along X and Y axes respectively with equal speeds of 10m/s. Find the initial velocity of the third part.

Solution 22

resultant velocity of 2 particles moving along x and y-axis

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m/s C.O.L.M

So,

V is along below x-axis.

Question 23 Two fat astronauts each of mass 120kg are travelling in a closed spaceship moving at a speed of 15km/s in the outer space far removed from all other material objects. The total mass of the spaceship and its contents including the astronauts is 660kg. If the astronauts do slimming exercise and thereby reduce their masses to 90kg each, with what velocity will the spaceship move?

Solution 23 In a closed spaceship, there is no external force (not even gravity). So the spaceship will move with a constant speed of 15m/s.

Question 24 During a heavy rain, hailstones of average size 1.0cm in diameter fall with an average speed of 20m/s. Suppose 2000 hailstorms strike every square meter of a 10m 10m roof perpendicularly in one second and the average force exerted by the falling hailstones on the roof. Density of a hailstone is 900.

Solution 24 Volume of 1 hailstorm

Mass= kg

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Mass of 2000 hailstorms= kg

average force on 1 roof

N Average force on 10m×10m (100 )roof

N

Question 25 A ball of mass m is dropped onto a floor from a certain height. The collision is perfectly elastic and the ball rebounds to the same height and again falls. Find the average force exerted by the ball on the floor during a long time interval.

Solution 25

For falling down, Time taken

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For bouncing up (back)

Question 26 A railroad car of mass M is at rest on frictionlessrails when a man of mass m starts moving on the car towards the engine. If the car recoils with a speed v backward on the rails, with what velocity is the man approaching the engine?

Solution 26 C.OL.M

velocity with which man in car approaches.

Question 27 A gun is mounted on a railroad car. The mass of the car, the gun, the shells and the operator is 50 m where m is the mass of one shell. If the velocity of the shell with respect to the gun (in its state before firing) is 200m/s, what is the recoil speed of the car after the second shot? Neglect the friction.

Solution 27 After 1 bullet, C.O.L.M

After 2nd bullet C.O.L.M

m/s (after 2nd shot)

Question 28 Two persons each of mass m are standing at the two extremes of a railroad car of mass M resting on a smooth track. The person on left jumps to the left with the horizontal speed u with respect to the state of car before the jump. Thereafter, the other person jumps to the right, again with the same

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horizontal speed u with respect to the state of car before the jump. Find the velocity of the car after both the persons have jumped off.

Solution 28 After left person jumps,

u is left, is right After right person jumps

u is left, is right (as is left)

Net velocity=

= towards left

Question 29 Figure shows a small block of mass m which is started with a speed of v on the horizontal part of the bigger block of mass M placed on a horizontal floor. The curved part of the surface shown is semicircular. All the surfaces are frictionless. Find the speed of the bigger block when the smaller block reaches the point A of the surface.

Solution 29 Use C.O.L.M

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Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 162 Question 30 In a typical Indian Bugghi (a luxury cart drawn by horses), a wooden plate is fixed on the rear on which one person can sit. A bugghiof mass 200kg is moving at a speed of 10km/h. As it overtakes a school boy walking at a speed of 4km/h, the boy sits on the wooden plate. If the mass of the boy is 25kg, what will be the new velocity of the bugghi?

Solution 30 C.O.L.M (conservation of linear momentum)

s-school boy, b-bugghi

m/s

Question 31 A ball of mass 0.50kg moving at a speed of 5.0m/s collides with another ball of mass 1.0kg. After the collision the balls stick together and remain motionless. What was the velocity of the 1.0kg block before the collision?

Solution 31 Use C.O.L.M

m/s

Question 32 A 60kg man skating with a speed of 10m/s collides with a 40kg skater at rest and they cling to each other. Find the loss of kinetic energy during the collision.

Solution 32

Use C.O.L.M

m/s

=1200J

Question 33 Consider a head-on collision between two particles of masses and . The initial speeds of the particles are and in the same direction. The collision starts at t=0 and the particles interact for a time interval . During the collision, the speed of the first particle varies as

Find the speed of the second particle as a function of time during the collision.

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Solution 33 Use C.O.L.M,

Question 34 A bullet of mass m moving at a speed v hits a ball of mass M kept at rest. A small part having

mass breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed of in the direction of bullet. Find the velocity of the bullet after the collision.

Solution 34

Use C.O.L.M,

Question 40 In a gamma decay process, the initial energy of a nucleus of mass M decreases, a gamma photon of energy E and linear momentum E/c is emitted and the nucleus recoils. Find the decrease in internal energy.

Solution 40

K.E. of nucleus= Decrease in internal energy

Question 41 A block of mass 2.0kg is moving on a frictionless horizontal surface with a velocity of 1.0m/s towards another block of mass kept at rest. The spring constant of the spring fixed at one end is 100N/m. Find the maximum compression of the spring.

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Solution 41 We know that for the spring,

Use law of kinematics,

m x=10cm

Question 42 A bullet of mass 20g travelling horizontally with a speed of 500m/s passes through a wooden block of mass 10.0kg initially at rest on a level surface. The bullet emerges with a speed of 100m/s and the block slides 20cm on the surface before coming to rest. Find the friction coefficient between the block and the surface.

Solution 42 FBD of block after collision

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Use C.O.L.M

m/s Use law of kinematics,

Question 43 A projectile is fired with a speed u at an angle above a horizontal field. The coefficient of restitution of collision between the projectile and the field is e. How far from the starting point, does the projectile makes its second collision with the field?

Solution 43

e=co-efficient of restitution

After collision with ground,

is the equation of trajectory.

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{y=0 at second projectile after it falls down from 1st projectile}

From starting point, it falls after distance.

Question 44 A ball falls on an inclined plane of inclination from a height h above the point of impact and makes a perfectly elastic collision. Where will it hit the plane again?

Solution 44

After falling down, ball will have a projectile motion.

Equation of trajectory

Question 46 A block of mass 200g is suspended through a vertical spring. The spring is stretched by 1.0cm when the block is in equilibrium. A particle of mass 120g is dropped on the block from a height of 45cm. The particle sticks to the block after the impact. Find the maximum extension of the spring. Take .

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Solution 46

At equilibrium

(before 120g falls) N/m

m/s Use C.O.L.M,

m/s If spring is stretched extra by According to conservation of energy,

cm

Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 163 Question 35 A ball of 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 three fourths of the original. Find the coefficient of restitution.

Solution 35

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Use C.O.L.M v=initial velocity

final velocities -(1)

Given

-(2) Solve (1) and (2)

Question 36 A block of mass 2.0kg moving at 2.0m/s collides head on with another block of equal mass kept at rest. (a) Find the maximum possible loss in kinetic energy due to the collision. (b) If the actual loss in kinetic energy is half of this maximum, find the co-efficient of restitution.

Solution 36 (a) Use C.O.L.M

For maximizing, take derivative.

m/s

J

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(b) v

Question 37 A particle of mass 100g moving at an initial speed u collides with another particle of same mass kept initially at rest. If the total kinetic energy becomes 0.2J after the collision, what could be the minimum and the maximum value of u.

Solution 37 Use C.O.L.M

J

When

m/s when or

if if as

m/s

Question 39 A ball falls on the ground from a height of 2.0m and rebounds up to a height of 1.5m. Find the coefficient of restitution.

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Solution 39

Question 45

Solve the previous problem if the coefficient of restitution is e. Use and .

Solution 45

and

=10m/s

m/s

Angle of reflection

Angle of projection

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l=18.5m

Question 47 A bullet of mass 25g is fired horizontally into a ballistic pendulum of mass 5.0kg and gets embedded in it. If the center of the pendulum rises by a distance of 10cm, find the speed of the bullet.

Solution 47 Use C.O.L.M

From C.O.E.L (conservation of energy law)

m/s

Question 48 A bullet of mass 20g moving horizontally at a speed of 300m/s is fired into a wooden block of mass 500g suspended by a long string. The bullet crosses the block and the block rises through a height of 20.0cm, find the speed of the bullet as it emerges from the block.

Solution 48 Use C.O.L.M

From C.O.E.L,

m/s

m/s

Question 49 Two masses and are connected by a spring of spring constant k and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance when the distance is released from rest. Find the distance moved by the two masses before they again come to rest.

Solution 49 Let and be travelled by and

{integrate}

Use C.O.E.L,

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Similarly,

Question 50 Two blocks of masses and are connected by a spring of spring constant k. The block of mass is given a sharp impulse so that it acquires a velocity towards right. Find (a) the velocity of the center of mass, (b) the maximum elongation that the spring will suffer.

Solution 50

(a) (b) Use C.O.E.L

Question 51 Consider the situation of the previous problem. Suppose each of the blocks is pulled by a constant force F instead of any impulse. Find the maximum elongation that the spring will suffer and the distances moved by the two blocks in the process.

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Solution 51 If and is travelled by under F Work done

Use C.O.E.L

Use C.O.L.M

Question 52 Consider the situation of the previous problem. Suppose the block of mass is pulled by a constant force and the other block is pulled by a constant force . Find the maximum elongation that the spring will suffer.

Solution 52

Net force of

Net force of

Use C.O.E.L,

Question 53 Consider a gravity-free hall in which a experimenter of mass 50kg is resting on a 5kg pillow, 8 ft above the floor of the hall. He pushes the pillow down so that it starts falling at the speed of 8 fts/s. The pillow

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makes a perfectly elastic collision with the floor, rebounds and reaches the experimenter's head. Find the time elapsed in the process.

Solution 53 ft/s

(velocity of pillow w.r.t man)

Use C.O.L.M,

{ ft/s}

ft/s

Time taken for going down=

sec Total time taken (up+ down) = sec

Question 54 The track shown in figure is frictionless. The block B of mass 2m is lying at rest and the block A of mass m is pushed along the track with speed. The collision between A and B is perfectly elastic. With what velocity should the block A be started to get the sleeping man awakened?

Solution 54 Use C.O.L.M,

When B reaches man, Use C.O.E.L,

Use C.O.L.M of whole system

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Solve (1) and (2) to get

Use

Block's initial velocity

Question 55 A bullet of mass 10g moving horizontally at a speed of m/s strikes a block of mass 490g kept on a frictionless track as shown in figure. The bullet remains inside the block and the system proceeds towards the semicircular track of radius 0.2m. Where will the block strike the horizontal part after leaving the semicircular track?

Solution 55

Use C.O.L.M,

m/s

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( =velocity at position B)

Use C.O.E.L

Angle of projection=

Distance in horizontal direction

m

Vertical direction distance=

Total distance= =0.22m

Question 56 Two balls having masses m and 2m are fastened to two light strings of same length l. The other ends of the strings are fixed at O. The strings are kept in the same horizontal line and the system is released from rest. The collision between the balls is elastic. (a) Find the velocities of the balls just after their collision. (b) How high will the balls rise after the collision?

Solution 56 Aak

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(a) For initial velocities, Use C.O.E.L

Use C.O.L.M

-(1) Since collision is elastic, e=1

-(2) Solve (1) and (2),

Similarly, (b) Use C.O.E.L

For mass m, use C.O.E.L,

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2m will rise up to,

m will rise up to,

Question 57 A uniform chain of mass M and length L is held vertically in such a way that its lower end just touches the horizontal floor. The chain is released from rest in this position. Any portion that strikes the floor comes to rest. Assuming that the chain does not form a heap on the floor, calculate the force exerted by it on the floor when a length x has reached the floor.

Solution 57 Consider Mass per unit length of chain

After chain's release,

Weight of chain due to x length

{M'=Mass of x length}

Total force=

Question 58 The blocks shown in figure have equal masses. The surface of A is smooth but that of B has a friction coefficient of 0.10 with the floor. Block A is moving at a speed of 10m/s towards B which is kept at rest. Find the distance travelled by B if (a) the collision is perfectly elastic and (b) the collision is perfectly inelastic. Take g=10 .

Solution 58 (a) Use C.O.L.M

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-(1)

Use C.O.E.L, -(2)

Solving (1) and (2)

If

Use kinematics law,

m (b) If it is inelastic, C.O.L.M

m/s

m

Question 59 The friction coefficient between the horizontal surface and each of the blocks is shown in figure is 0.20. The collision between the blocks is perfectly elastic. Find the separation between the two blocks when they come to rest. Take .

Solution 59 Using work energy principle V=velocity of 2kg near collision

m/s Use C.O.L.M, { =final velocities}

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- (1) As collision is elastic, e=1

- (2) Solving (1) and (2)

m/s

m/s Use C.O.E.L for 1st block,

cm

Use C.O.E.L for 2nd block,

cm

Distance between them=4+1=5cm

Question 60 A block of mass m is placed on a triangular block of mass M, which in turn is placed on a horizontal surface as shown in figure. Assuming frictionless surfaces between the velocities of the triangular block when the smaller block reaches the bottom end.

Solution 60 Along x-direction

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=resultant

Vertical distance travelled by m=

Use law of kinematics,

{velocity of M block}

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Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 164 Question 61 Figure shows a small body of mass m placed over a larger mass M whose surface is horizontal near the smaller mass and gradually curves to become vertical. The smaller mass is pushed on the larger ones at a speed v and the system is left to itself. Assume that all the surfaces are frictionless. (a) Find the speed of the larger block when the smaller block is sliding on the vertical part. (b) Find the speed of the smaller mass when it breaks off the larger mass at height h. (c) Find the maximum height (from the ground) that the smaller mass ascends. (d) Show that the smaller mass will again land on the bigger one. Find the distance traversed by the bigger block during the time when the smaller block was in its flight under gravity.

Solution 61 (a) Use C.O.L.M

(b) Use C.O.E.L

(c) {resultant velocity}

Use C.O.E.L

Total height=

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= (d) time for flight

Total time=

Question 62 A small block of superdense material has a mass of kg. It is situated at a height h (much smaller than the earth's surface) from where it falls on the earth's surface. Find its speed when its height from the earth's surface has reduced to h/2. The mass of the earth is kg.

Solution 62 b-block e-earth Use C.O.L.M

Use C.O.E.L

h<<R

Question 63 A body of mass m makes an elastic collision with another identical body at rest. Show that if the collision is not head-on, the bodies go right angle to each other after the collision.

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Solution 63

If m is not colliding head-on, along x-axis, use C.O.L.M

-(1) Along y-axis, -(2) Use C.O.E.L

Squaring (1) and (2) and add them,

Question 64 A small particle travelling with a velocity v collides elastically with a spherical body of equal mass and of radius r initially kept at rest. The center of this spherical body is located a distance ( ) away from the direction of motion of the particle. Find the final velocities of the two particles.

[Hint:- The force acts along the normal to the sphere through the contact. Treat the collision as one-dimensional for this direction. In the tangential direction no force acts and velocities do not change.]

Solution 64

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{after collision}

Chapter 9 - Centre of Mass, Linear Momentum, Collision Exercise 169 Question 13 A cart of mass M is at rest on a frictionless horizontal surface and a pendulum bob of mass m hangs from the roof of the cart. The string breaks, the bob falls on the floor, makes several collisions on the floor and finally lands up in a small slot made in the floor. The horizontal distance between the string and the slot is L. Find the displacement of the cart during this process.

Solution 13

Take as origin

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Cart should be displaced cm for COM of system to be same.

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