Concepts of Physics

H. C. Verma

Chapter 9

Centre of Mass, Linear Momentum, Collision - all with Video Answers

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Chapter Questions

Problem 1

Three particles of masses $1 \cdot 0 \mathrm{~kg}, 2 \cdot 0 \mathrm{~kg}$ and $3 \cdot 0 \mathrm{~kg}$ are placed at the corners $A, B$ and $C$ respectively of an equilateral triangle $A B C$ of edge $1 \mathrm{~m}$. Locate the centre of mass of the system.

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Problem 2

The structure of a water molecule is shown in figure (9-E1). Find the distance of the centre of mass of the molecule from the centre of the oxygen atom.

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Problem 3

Seven homogeneous bricks, each of length $L$, are arranged as shown in figure (9-E2). Each brick is displaced with respect to the one in contact by $L / 10$. Find the $x$ -coordinate of the centre of mass relative to the origin shown.

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Problem 4

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

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Problem 5

A disc of radius $R$ is cut out from a larger disc of radius $2 R$ in such a way that the edge of the hole touches the edge of the disc. Locate the centre of mass of the residual disc.

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Problem 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 (9-Q2). Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.

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Problem 7

Calculate the velocity of the centre of mass of the system of particles shown in figure $(9-\mathrm{E} 3)$.

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Problem 8

Two blocks of masses $10 \mathrm{~kg}$ and $20 \mathrm{~kg}$ are placed on the $X$ -axis. The first mass is moved on the axis by a distance of $2 \mathrm{~cm} .$ By what distance should the second mass be moved to keep the position of the centre of mass unchanged ?

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Problem 9

Two blocks of masses $10 \mathrm{~kg}$ and $30 \mathrm{~kg}$ are placed along a vertical line. The first block is raised through a height of $7 \mathrm{~cm} .$ By what distance should the second mass be moved to raise the centre of mass by $1 \mathrm{~cm}$ ?

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Problem 10

Consider a gravity-free hall in which a tray of mass $M$, carrying a cubical block of ice of mass $m$ and edge $L$, is at rest in the middle (figure 9 -E4). If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descend?

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Problem 11

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ (figure 9-E5).

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Problem 12

Mr. Verma ( $50 \mathrm{~kg}$ ) and Mr. Mathur ( $60 \mathrm{~kg}$ ) are sitting at the two extremes of a $4 \mathrm{~m}$ long boat ( $40 \mathrm{~kg}$ ) standing still in water. To discuss a mechanics 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?

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Problem 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 (figure 9-E6). 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.

Naresh Bagrecha

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Problem 14

The balloon, the light rope and the monkey shown in figure $(9-\mathrm{E} 7)$ are at rest in the 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 the length of the rope ascended by the monkey $=L$

Naresh Bagrecha

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Problem 15

Find the ratio of the linear momenta of two particles of masses $1 \cdot 0 \mathrm{~kg}$ and $4 \cdot 0 \mathrm{~kg}$ if their kinetic energies are equal.

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Problem 16

A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of $1 \cdot 4 \times 10^{7} \mathrm{~m} / \mathrm{s}$. Calculate the recoil speed of the residual nucleus thorium-234. Assume that the mass of a nucleus is proportional to the mass number.

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Problem 17

A man of mass $50 \mathrm{~kg}$ starts moving on the earth and acquires a speed of $1 \cdot 8 \mathrm{~m} / \mathrm{s}$. With what speed does the earth recoil? Mass of earth $=6 \times 10^{24} \mathrm{~kg}$.

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Problem 18

A neutron initially at rest, decays into a proton, an electron and an antineutrino. The ejected electron has a momentum of $1 \cdot 4 \times 10^{-26} \mathrm{~kg}-\mathrm{m} / \mathrm{s}$ and the antineutrino $6 \cdot 4 \times 10^{-27} \mathrm{~kg}-\mathrm{m} / \mathrm{s}$. 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 $=1 \cdot 67 \times 10^{-27} \mathrm{~kg}$

Naresh Bagrecha

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Problem 19

A man of mass $M$ having a bag of mass $m$ slips from the roof of a tall building of height $H$ and starts falling vertically (figure 9 -E8). When at a height $h$ from the ground, he notices that the ground below him is pretty hard, but there is a pond at a horizontal distance $x$ from the line of fall. In order to save himself he throws the bag horizontally (with respect to himself) in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in the water. If the man just succeeds to avoid the hard ground, where will the bag land?

Naresh Bagrecha

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Problem 20

A ball of mass $50 \mathrm{~g}$ moving at a speed of $2 \cdot 0 \mathrm{~m} / \mathrm{s}$ strikes a plane surface at an angle of incidence $45^{\circ}$. 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) the change in the magnitude of the momentum of the ball.

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Problem 21

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

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Problem 22

A block at rest explodes into three equal parts. Two parts start moving along $X$ and $Y$ axes respectively with equal speeds of $10 \mathrm{~m} / \mathrm{s}$. Find the initial velocity of the third part.

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Problem 23

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

Naresh Bagrecha

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Problem 24

During a heavy rain, hailstones of average size $1 \cdot 0 \mathrm{~cm}$ in diameter fall with an average speed of $20 \mathrm{~m} / \mathrm{s}$. Suppose 2000 hailstones strike every square meter of a $10 \mathrm{~m} \times 10 \mathrm{~m}$ 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 a hailstone is $900 \mathrm{~kg} / \mathrm{m}^{3}$.

Naresh Bagrecha

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Problem 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.

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Problem 26

A railroad car of mass $M$ is at rest on frictionless rails 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?

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Problem 27

A gun is mounted on a railroad car. The mass of the car, the gun, the shells and the operator is $50 \mathrm{~m}$ 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 $200 \mathrm{~m} / \mathrm{s}$, what is the recoil speed of the car after the second shot? Neglect friction.

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Problem 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 (figure 9-E10). The person on left jumps to the left with a horizontal speed $u$ with respect to the state of the car before the jump. Thereafter, the other person jumps to the right, again with the same horizontal speed $u$ with respect to the state of the car before his jump. Find the velocity of the car after both the persons have jumped off.

Naresh Bagrecha

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Problem 29

Figure (9-E11) shows a small block of mass $m$ which is started with a speed $v$ 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.

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Problem 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 bugghi of mass $200 \mathrm{~kg}$ is moving at a speed of $10 \mathrm{~km} / \mathrm{h}$. As it overtakes a school boy walking at a speed of $4 \mathrm{~km} / \mathrm{h}$, the boy sits on the wooden plate. If the mass of the boy is $25 \mathrm{~kg}$, what will be the new velocity of the bugghi?

Naresh Bagrecha

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Problem 31

A ball of mass $0: 50 \mathrm{~kg}$ moving at a speed of $5 \cdot 0 \mathrm{~m} / \mathrm{s}$ collides with another ball of mass $1 \cdot 0 \mathrm{~kg}$. After the collision the balls stick together and remain motionless. What was the velocity of the $1 \cdot 0 \mathrm{~kg}$ block before the collision?

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Problem 32

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

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Problem 33

Consider a head-on collision between two particles of masses $m_{1}$ and $m_{2}$. The initial speeds of the particles are $u_{1}$ and $u_{2}$ in the same direction. The collision starts at $t=0$ and the particles interact for a time interval $\Delta t$. During the collision, the speed of the first particle varies as
$$
v(t)=u_{1}+\frac{t}{\Delta t}\left(v_{1}-u_{1}\right)
$$
Find the speed of the second particle as a function of time during the collision.

Naresh Bagrecha

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Problem 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 $m^{\prime}$ breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed $v_{1}$ in the direction of the bullet. Find the velocity of the bullet after the collision.A bullet of mass $m$ moving at a speed $v$ hits a ball of mass $M$ kept at rest. A small part having mass $m^{\prime}$ breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed $v_{1}$ in the direction of the bullet. Find the velocity of the bullet after the collision.

Naresh Bagrecha

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Problem 35

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 three fourths of the original. Find the coefficient of restitution.

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Problem 36

A block of mass $2 \cdot 0 \mathrm{~kg}$ moving at $2 \cdot 0 \mathrm{~m} / \mathrm{s}$ 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 coefficient of restitution.

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Problem 37

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

Naresh Bagrecha

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Problem 38

Two friends $A$ and $B$ (each weighing $40 \mathrm{~kg}$ ) are sitting on a frictionless platform some distance $d$ apart. $A$ rolls a ball of mass $4 \mathrm{~kg}$ on the platform towards $B$ which $B$ catches. Then $B$ rolls the ball towards $A$ and $A$ catches it. The ball keeps on moving back and forth between $A$ and $B$. The ball has a fixed speed of $5 \mathrm{~m} / \mathrm{s}$ on the platform. (a) Find the speed of $A$ after he rolls the ball for the first time. (b) Find the speed of $A$ after he catches the ball for the first time. (c) Find the speeds of $A$ and $B$ after the ball has made 5 round trips and is held by
A. (d) How many times can $A$ roll the ball ? (e) Where is the centre of mass of the system $" A+B+$ ball" at the end of the $n$ th trip?

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Problem 39

A ball falls on the ground from a height of $2 \cdot 0 \mathrm{~m}$ and rebounds up to a height of $1 \cdot 5 \mathrm{~m}$. Find the coefficient of restitution.

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Problem 40

In a gamma decay process, the internal 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.

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Problem 41

A block of mass $2 \cdot 0 \mathrm{~kg}$ is moving on a frictionless horizontal surface with a velocity of $1 \cdot 0 \mathrm{~m} / \mathrm{s}$ (figure 9 -E12) towards another block of equal mass kept at rest. The spring constant of the spring fixed at one end is 100 $\mathrm{N} / \mathrm{m} .$ Find the maximum compression of the spring.

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Problem 42

A bullet of mass $20 \mathrm{~g}$ travelling horizontally with a speed of $500 \mathrm{~m} / \mathrm{s}$ passes through a wooden block of mass $10^{\circ} 0$ kg initially at rest on a level surface. The bullet emerges with a speed of $100 \mathrm{~m} / \mathrm{s}$ and the block slides $20 \mathrm{~cm}$ on the surface before coming to rest. Find the friction coefficient between the block and the surface (figure 9-E13).

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Problem 43

A projectile is fired with a speed $u$ at an angle $\theta$ 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?

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Problem 44

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

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Problem 45

Solve the previous problem if the coefficient of restitution is $e$. Use $\theta=45^{\circ}, e=\frac{3}{4}$ and $h=5 \mathrm{~m}$.

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Problem 46

A block of mass $200 \mathrm{~g}$ is suspended through a vertical spring. The spring is stretched by $1 \cdot 0 \mathrm{~cm}$ when the block is in equilibrium. A particle of mass $120 \mathrm{~g}$ is dropped on the block from a height of $45 \mathrm{~cm}$. The particle sticks to the block after the impact. Find the maximum extension of the spring. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

Naresh Bagrecha

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Problem 47

A bullet of mass $25 \mathrm{~g}$ is fired horizontally into a ballistic pendulum of mass $5 \cdot 0 \mathrm{~kg}$ and gets embedded in it (figure 9-E14). If the centre of the pendulum rises by a distance of $10 \mathrm{~cm}$, find the speed of the bullet.

Naresh Bagrecha

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Problem 48

A bullet of mass 20 g moving horizontally at a speed of $300 \mathrm{~m} / \mathrm{s}$ is fired into a wooden block of mass $500 \mathrm{~g}$ suspended by a long string. The bullet crosses the block and emerges on the other side. If the centre of mass of the block rises through a height of $20 \cdot 0 \mathrm{~cm}$, find the speed of the bullet as it emerges from the block.

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Problem 49

Two masses $m_{1}$ and $m_{2}$ 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 $x_{0}$ when the system is released from rest. Find the distance moved by the two masses before they again come to rest.

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Problem 50

Two blocks of masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$ (figure 9-E15). The block of mass $m_{2}$ is given a sharp impulse so that it acquires a velocity $v_{0}$ towards right. Find (a) the velocity of the centre of mass, (b) the maximum elongation that the spring will suffer.

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Problem 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.

Naresh Bagrecha

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Problem 52

Consider the situation of the previous problem. Suppose the block of mass $m_{1}$ is pulled by a constant force $F_{1}$ and the other block is pulled by a constant force $F_{2}$. Find the maximum elongation that the spring will suffer.

Naresh Bagrecha

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Problem 53

Consider a gravity-free hall in which an experimenter of mass $50 \mathrm{~kg}$ is resting on a $5 \mathrm{~kg}$ pillow, $8 \mathrm{ft}$ above the floor of the hall. He pushes the pillow down so that it starts falling at a speed of $8 \mathrm{ft} / \mathrm{s}$. The pillow makes a perfectly elastic collision with the floor, rebounds and reaches the experimenter's head. Find the time elapsed in the process.

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Problem 54

The track shown in figure (9-E16) is frictionless. The block $B$ of mass $2 m$ is lying at rest and the block $A$ of mass $m$ is pushed along the track with some 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?

Naresh Bagrecha

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Problem 55

A bullet of mass 10 g moving horizontally at a speed of $50 \sqrt{7 \mathrm{~m} / \mathrm{s}}$ strikes a block of mass $490 \mathrm{~g}$ kept on a frictionless track as shown in figure (9-E17). The bullet remains inside the block and the system proceeds towards the semicircular track of radius $0 \cdot 2 \mathrm{~m}$. Where will the block strike the horizontal part after leaving the semicircular track?

Steven Brown

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Problem 56

Two balls having masses $m$ and $2 m$ are fastened to two light strings of same length $l$ (figure 9-E18). 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?

Stephen Zaffke

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Problem 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.

Stephen Zaffke

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Problem 58

The blocks shown in figure (9-E19) have equal masses. The surface of $A$ is smooth but that of $B$ has a friction coefficient of $0 \cdot 10$ with the floor. Block $A$ is moving at a speed of $10 \mathrm{~m} / \mathrm{s}$ 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 $g$ $=10 \mathrm{~m} / \mathrm{s}^{2}$

Stephen Zaffke

Numerade Educator

Problem 59

The friction coefficient between the horizontal surface and each of the blocks shown in figure (9-E20) is $0 \cdot 20$. The collision between the blocks is perfectly elastic. Find the separation between the two blocks when they come to rest. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

Stephen Zaffke

Numerade Educator

Problem 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 (9-E21). Assuming frictionless surfaces find the velocity of the triangular block when the smaller block reaches the bottom end.

Manish Jain

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Problem 61

Figure (9-E22) 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 longer one 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.

Manish Jain

Numerade Educator

Problem 62

A small block of superdense material has a mass of $3 \times 10^{24} \mathrm{~kg} .$ It is situated at a height $h$ (much smaller than the earth's radius) 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 $6 \times 10^{24} \mathrm{~kg}$

Stephen Zaffke

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Problem 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 at right angle to each other after the collision.

Manish Jain

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Problem 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 centre of this spherical body is located a distance $\rho(<r)$ away from the direction of motion of the particle (figure 9-E23). Find the final velocities of the two particles.

Manish Jain

Numerade Educator