Can the center of mass of a body be at a point outside the

body?

If all the particles of a system lie in X-Y plane, is it necessary that the center of mass be in X-Y plane?

If all the particles of a system lie in a cube, is it necessary that the center of mass be in the cube?

The center of mass is defined as Suppose we define “center of charge” as where q_i represents the ith charge placed at and Q is the total charge of the system.

(a) Can the center of charge of a two-charge system be outside the line segment joining the charges?

(b) If all the charges of a system are in X.Y plane, is it necessary that the center of charge be in X-Y plane?

(c) If all the charges of a system lie in a cube, is it necessary that the center of charge be in the cube?

The weight Mg of an extended body is generally shown in a diagram to act through the center of mass. Does it mean that the earth does not attract other particles?

A bob suspended from the ceiling of a car which is accelerating on a horizontal road. The bob stays at rest with respect to the car with the string making an angle e with the vertical. The linear momentum of the bob as seen from the road is increasing with time. Is it a violation of conservation of linear momentum? If not, where is the external force which changes the linear momentum?

You are waiting for a train on a railway platform. Your three-year-old niece is standing on your iron trunk containing the luggage, why does the trunk not recoil as she jumps off on the platform?

In a head-on collision between two particles, is it necessary that the particles will acquire a common velocity at least for one instant?

A collision experiment is done on a horizontal table kept in an elevator. Do you expect a change in the results if the elevator is accelerated up or down because of the non-inertial character of the frame?

Two bodies make an elastic head-on collision on a smooth horizontal table kept in a car. Do you expect a change in the result if the car is accelerated on a horizontal road because of the non-inertial character of the frame? Does the equation “Velocity of separation = Velocity of approach” remain valid in an accelerating car? Does the equation “final momentum = initial momentum” remain valid in the accelerating car?

If the total mechanical energy of a particle is zero, is its linear momentum necessarily zero? Is it necessarily nonzero?

If the linear momentum of a particle is known, can you find its kinetic energy? if the kinetic energy of a particle is known can you find its linear momentum?

What can be said about the center of mass of a uniform hemisphere without making any calculation? Will its distance from the center be more than r/2 or less than r/2?

You are holding a cage containing a bird. Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage? Does it make a difference whether the cage is completely closed or it has rods to let air pass?

A fat person is standing on a light plank floating on a calm lake. The person walks from one end to the other on the plank. His friend sitting on the shore watches him and finds that the person hardly moves any distance because the plank moves backward about the same distance as the person moves on the plank. Explain.

A high-jumper successfully clears the bar. Is it possible that his center of mass crossed the bar from below it? Try it with appropriate figures.

Which of the two persons shown in figure (9-Q 1) is more likely to fait down? Which external force is responsible for his falling down?

Suppose we define a quantity ‘Linear Momentum’ as

linear momentum = mass x speed.

The linear momentum of a system of particles is the sum of linear momenta of the individual particles. Can we state a principle of conservation of linear momentum as “linear momentum of a system remains constant if no external force acts on it”?

Use the definition of linear momentum from the previous question. Can we state the principle of conservation of linear momentum for a single particle?

To accelerate a car we ignite petrol in the engine of the car. Since only an external force can accelerate the center of mass, is it proper to say that “the force generated by the engine accelerates the car”?

A ball is moved on a horizontal table with some velocity. The ball stops after moving some distance. Which external force is responsible for the change in the momentum of the ball?

Consider the situation of the previous problem. Take “the table plus the ball” as the system. Friction between the table and the ball is then an internal force. As the ball slows down, the momentum of the system decreases. Which external force is responsible for this change in momentum?

When a nucleus at rest emits a beta particle, it is found that the velocities of the recoiling nucleus and the beta particle are not along the same straight line. How can this be possible in view of the principle of conservation of momentum?

A van is standing on a frictionless portion of a horizontal road. To start the engine, the vehicle must be set in motion in the forward direction. How can the persons sitting inside the van do it without corning out and pushing from behind?

In one-dimensional elastic collision of equal masses, the velocities are interchanged. Can velocities in a one-dimensional collision be interchanged if the masses are not equal?

Consider the following two equations:

(A)

and (B)

In a noninertial frame

Consider the following two statements:

(A) Linear momentum of the system remains constant.

(B) Centre of mass of the system remains at rest.

Consider the following two statements:

(A) Linear momentum of a system of particles is zero.

(B) Kinetic energy of a system of particles is zero.

Consider the following two statements:

(A) The linear momentum of a particle is independent of the frame of reference.

(B) The kinetic energy of a particle is independent of the frame of reference.

All the particles of a body are situated at a distance R from the origin. The distance of the centre of mass of the body from the origin is

A circular plate of diameter a is kept in contact with a square plate of edge a as shown in figure (9.Q2). The density of the material and the thickness are same

everywhere. The centre of mass of the composite system will be

Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration The center of mass has an acceleration

Internal forces can change

A bullet hits a block kept at rest on a smooth horizontal surface and gets embedded into it. Which of the following does not change?

A uniform sphere is placed on a smooth horizontal surface and a horizontal force F is applied on it at a distance h above the surface. The acceleration of the centre

A body falling vertically downwards under gravity breaks in two parts of unequal masses. The centre of mass of the two parts taken together shifts horizontally towards

A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass

A body at rest breaks into two pieces of equal masses. The parts will move

A heavy ring of mass m is damped on the periphery of a light circular disc. A small particle having equal mass is clamped at the centre of the disc. The system is rotated in such a way that the centre moves in a circle of radius r with a uniform speed υ. We conclude that an external force

The quantities remaining constant in a collision are

A nucleus moving with a velocity emits an α-particle. Let the velocities of the α-particle and the remaining nucleus be and , and their masses be m₁ and m₂.

A shell is fired from a cannon with a velocity V at an angle θ with the horizontal direction. At the highest point in its path, it explodes into two pieces of equal masses. One of the pieces retraces its path to the cannon. The speed of the other piece immediately after the explosion is

In an elastic collision

In an inelastic collision

The centre of mass of a system of particles is at the origin. It follows that

A body has its centre of mass at the origin. The x-coordinates of the particles

In which of the following cases the centre of mass of a rod is certainly not at its centre?

If the external forces acting on a system have zero resultant, the centre of mass

A nonzero external force acts on a system of particles. The velocity and the acceleration of the centre of mass are found to be u₀ and a₀ at an instant t. It is possible that

Two balls are thrown simultaneously in air. The acceleration of the centre of mass of the two balls while in air

A block moving in air breaks in two parts and the parts separate

In an elastic collision

A ball hits a floor and rebounds after an inelastic collision. In this case

A body moving towards a finite body at rest collides with it. It is possible that

In a head-on elastic collision of two bodies of equal masses

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

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

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 center of mass relative to the origin shown.

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.

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.

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 center of mass of the combination, assuming same mass per unit area for the two plates.

Calculate the velocity of the centre of mass of the system of particles shown in figure (9-E3).

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

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

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?

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R₁ and R₂ (figure 9-E5).

Mr. Verma (50 kg) and Mr. Mathur (60 kg) are sitting at the two extremes of a 4 m long boat (40 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?

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 fly 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.

The balloon, the light rope and the monkey shown in figure (9-E7) 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.

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

A uranium-238 nucleus, initially at rest, emits an alpha particle with a speed of 1.4 × 10⁷ 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.

A man of mass 50 kg starts moving on the earth and acquires a speed of 1.8 m/s. With what speed does the earth recoil? Mass of earth = 6 × 10²⁴ kg.

A neutron initially at rest, decays into a proton, all electron and an antineutrino. The ejected electron has a momentum of 14 × 10^-28 kg-m/s and the antineutrino
6.4 × 10^-27 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 = 1.67 × X 10^-27 kg.

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?

A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence 45^o. 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.

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

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

Two fat astronauts each of mass 120 kg are travelling in a closed spaceship moving at a speed of 15 km/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 kg. If the astronauts do slimming exercise and thereby reduce their masses to 90 kg each, with what velocity will the spaceship move?

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 × 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 a hailstone is 900 kg/m³.

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.

A railroad car of mass M is at rest on frictionless rails when a man of mass in 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?

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 200 m/s, what is the recoil speed of the car after the second shot? Neglect friction.

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.

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.

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

Ans.

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

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

Consider a head-on collision between two particles of masses m₁ and m₂. The initial speeds of the particles are u₁ and u₂ in the same direction. The collision starts at t = 0 and the particles interact for a time interval ∆t. During the collision, the speed of the first particle vanes as

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

A bullet of mass m moving at a speed u hits a ball of mass M kept at rest. A small part having mass m’ breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed u₁ in the direction of the bullet. Find the velocity of the bullet after the collision.

A ball of mass m moving at a speed u 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.

A block of mass 2.0 kg moving at 2.0 m/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.

A particle of mass 100 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 J after the collision, what could be the minimum and the maximum value of u.

Two friends A and B (each weighing 40 kg) are sitting on a frictionless platform some distance d apart. A rolls a ball of mass 4 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 m/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 center of mass of the system “A + B + ball” at the end of the nth trip?

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

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.

A block of mass 2.0 kg is moving on a frictionless horizontal surface with a velocity of 1.0 m/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 N/m. Find the maximum compression of the spring.

A bullet of mass 20 g travelling horizontally with a speed of 500 m/s passes through a wooden block of mass 10.0 kg initially at rest on a level surface. The bullet emerges with a speed of 100 m/s and the block slides 20 cm on the surface before
coming to rest. Find the friction coefficient between the block and the surface (figure 9-E13).

A projectile is fired with a speed u at an angle θ 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?

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

Solve the previous problem if the coefficient of restitution is e. Use θ = 45°, and h = 5 m.

A block of mass 200 g is suspended through a vertical spring. The spring is stretched by 1.0 cm when the block is in equilibrium. A particle of mass 120 g is dropped on the block from a height of 45 cm. The particle sticks to the block after the impact. Find the maximum extension of the spring. Take g = 10 m/s².

A bullet of mass 25 g is fired horizontally into a ballistic pendulum of mass 5.0 kg and gets embedded in it (figure9.E14). If the center of the pendulum rises by a distance of 10 cm, find the speed of the bullet.

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

Two masses m₁ and m₂ 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 x0 when the system is released from rest. Find the distance moved by the two masses before they again come to rest.

Two blocks of masses m₁ and m₂ are connected by a spring of spring constant k (figure 9-E15). The block of mass m₂ is given a sharp impulse so that it acquires a velocity u₀ towards right. Find (a) the velocity of the center of mass, (b) the maximum elongation that the spring will suffer.

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.

Consider the situation of the previous problem. Suppose the block of mass m₁ is pulled by a constant force F₁, and the other block is pulled by a constant force F₂. Find the maximum elongation that the spring will suffer.

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

The track shown in figure (9-E16) 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 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?

A bullet of mass 10 g moving horizontally at a speed of 50√ m/s strikes a block of mass 490 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 0.2 m. Where will the block strike the horizontal part after leaving the semicircular track?

Two balls having masses m and 2m are fastened to two light strings of same length l (figure 9-E 18). 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?

A uniform chain of mass M and length L is held vertically in such a way that its lower endjust 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.

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 0.10 with the floor. Block A is moving
at a speed of 10 m/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 m/s².

The friction coefficient between the horizontal surface and each of the blocks shown in figure (9-E20) is 0.20. The collision between the blocks is perfectly elastic. Find the separation between the two blocks when they come to rest.
Take g = 10 m/s².

A block of mass m is placed on 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.

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 u 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.

A small block of super dense material has a mass of 3 × 10²⁴ 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 × 10²⁴ kg.

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.

A small particle travelling with a velocity υ 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 ρ(< r) away from the direction of motion of the particle (figure
9-E23). 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 the velocities do not change].