b)

### Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V as shown:If the collision is elastic, which of the following is a possible result after collision?

\b)

### Which of the following does not describe the elastic collision of two billiard balls? Distance between the centres of the balls is r.

(i), (ii), (iii), (iv) and (vi). The distance between two masses in a system is inversely proportional to their potential energy. As the two balls grow closer to each other, the potential energy of...

### On a frictionless track, a trolley moves with a speed of with a mass of . A child whose mass is 20 kg runs on the trolley with a speed of from one end to other which is . The speed is relative to the trolley in the direction opposite to its motion. Find the final speed of the trolley and the distance the trolley moved from the time the child began to run.

Mass is given as $m=200 \mathrm{Kg}$ Speed is given as $v=36 \mathrm{~km} / \mathrm{h}=10 \mathrm{~m} / \mathrm{s}$ Mass of boy is given as $=20 \mathrm{Kg}$ Initial momentum will be, $(M+m) v$...

### A bolt of mass falls from the ceiling of an elevator moving down with a uniform speed of . It hits the floor of the elevator (length of elevator ) and does not rebound. What is the heat produced by the impact? Would your answer be different if the elevator were stationary?

Mass of the bolt is given as $m=0.3 \mathrm{~kg}$ Potential energy of the bolt is given as $m g h=0.3 \times 9.8 \times 3=8.82\rfloor$ The bolt does not return to its original position. As a result,...

### A block situated on a rough incline is connected to a spring of spring constant 100 as shown in Fig. The block is released from rest with the spring in the unstretched position. The block moves down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Solution: Spring constant is given as $\mathrm{k}=100 \mathrm{~N} \mathrm{~m}^{m}$ Displacement in the block is given as $\mathrm{x}=10 \mathrm{~cm}=0.1 \mathrm{~m}$ At equilibrium: Normal reaction...

### Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track Fig. Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given , , and , what are the speeds and times taken by the two stones?

Solution: The sides $A B$ and $A C$ of the figure are both inclined to the horizontal at $\theta_{1}$ and $\theta_{2}$, respectively. According to the law of mechanical energy conservation,...

### A bullet of mass and horizontal speed strikes a block of wood of mass and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.

Mass of the bullet is given as $m_{1}=0.012 \mathrm{~kg}$ Initial speed of the bullet is given as $u_{1}=70 \mathrm{~m} / \mathrm{s}$ Mass of the wooden block is given as $m_{2}=0.4 \mathrm{~kg}$...

### A family uses of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of per square meter. If of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW?(b) Compare this area to that of the roof of a typical house.

(a) Power used by family is given as $p=8 \mathrm{KW}=8000 \mathrm{~W}$ Solar energy received per square metre is given as $200 \mathrm{~W} / \mathrm{m}^{2}$ Percentage of energy converted to useful...

### A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much work does she do against the gravitational force? (b) Fat supplies of energy per kilogram which is converted to mechanical energy with a efficiency rate. How much fat will the dieter use up?

Mass is given as $\mathrm{m}=10 \mathrm{~kg}$ Height to which the mass is lifted is given as $h=0.5 \mathrm{~m}$ Number of times is hiven as $n=1000$ (a) Work done against gravitational force can be...

### The windmill sweeps a circle of area A with their blades. If the velocity of the wind is perpendicular to the circle, find the air passing through it in time and also the kinetic energy of the air. of the wind energy is converted into electrical energy and and the density of the air is What is the electrical power produced?

Area = A Velocity $=\mathrm{V}$ Density $=\rho$ (a) Volume of the wind through the windmill per sec is given by $=\mathrm{Av}$ Mass is given by $=\rho \mathrm{AV}$ So, Mass $m$ through the windmill...

### A body of mass travels in a straight line with velocity where . What is the work done by the net force during its displacement from to m?

Let the mass of the body be $m$ $m=0.5 \mathrm{~kg}$ Velocity of the body is represented by $v=a x^{3 / 2}$ where, $a=5 \mathrm{~m}^{-1 / 2} \mathrm{~s}^{-1}$. Initial velocity at $x=0$ will be...

### A trolley of mass carrying a sandbag of is moving uniformly with a speed of on a frictionless track. After a while, the sand starts leaking out of a hole on the floor of the trolley at the rate of What is the speed of the trolley after the entire sandbag is empty?

The sandbag is placed in the trolley, which travels at a constant speed of 27 km/h. There is no system that acts as an external force. There will be no external force operating on the system even if...

### The bob of a pendulum is released from a horizontal position. If the length of the pendulum is , what is the speed with which the bob arrives at the lowermost point, given that it dissipated of its initial energy against air resistance?

Length of the pendulum is given as $\mid=1.5 \mathrm{~m}$ Potential of the bob at the horizontal position is given as $=m g h=m g \mid$ When the bob goes from the horizontal position to the lowest...

### A ball A which is at an angle to the vertical is released and it hits a ball B of same mass which is at rest. Does the ball A rises after collision? The collision is an elastic collision.

When the ball A collides with the stationary ball B in an elastic collision, the ball B gains the velocity of the ball A, while the ball A comes to a stop immediately after the collision....

### Two identical ball bearings in contact with each other and resting on a frictionless table is hit head-on by another ball bearing of the same mass moving initially with a speed . If the collision is elastic, which of the following figure is a possible result after collision?

Solution: The mass of the ball bearing is given as $\mathrm{m}$ Before the collision, Total Kinetic Energy of the system will be $=1 / 2 m v^{2}+0=1 / 2 m v^{2}$ After the collision, Total Kinetic...

### A pump on the ground floor of a building can pump up water to fill a tank of volume 30 in . If the tank is above the ground, and the efficiency of the pump is , how much electric power is consumed by the pump?

Volume of the tank is given as $30 \mathrm{~m}^{3}$ Time taken to fill the tank is given as $15 \mathrm{~min}=15 \times 60=900 \mathrm{~s}$ Height of the tank above the ground is given as $h=40...

### A molecule in a gas container hits a horizontal wall with speed and angle with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?

For an elastic or inelastic collision, momentum is always preserved. The molecule approaches and rebounds with the same speed of $200 \mathrm{~m} / \mathrm{s}$. $u=v=200 \mathrm{~m}...

### A raindrop of radius falls from a height of above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is ?

Radius of the drop is given as $2 \mathrm{~mm}=2 \times 10^{-3} \mathrm{~m}$. Height from which the raindrops fall is given as $\mathrm{S}=500 \mathrm{~m}$. The density of water is given as...

### An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy , and the second with . Which is faster, the electron or the proton? Obtain the ratio of their speeds. (electron mass , proton mass )

Electron mass is given as $m_{e}=9.11 \times 10^{-31} \mathrm{~kg}$ Proton mass is given as $m_{p}=1.67 \times 10^{-27} \mathrm{~kg}$ Electron's kinetic energy can be calculated as...

### A body constrained to move along the z-axis of a coordinate system is subject to a constant force given by where i, , are unit vectors along the – and -axis of the system respectively. What is the work done by this force in moving the body at a distance of along the z-axis?

The body is displaced by $4 \mathrm{~m}$ along $z$-axis, so we have, $\vec{S}=0 \hat{i}+0 \hat{j}+4 \hat{k}$ $\vec{F}=-\hat{i}+2 \hat{j}+3 \hat{k}$ Work done can be calculated as, $W=\vec{F} \cdot...

### A body is moving unidirectionally under the influence of a source of constant power. Its dis placement in time t is proportional to

(i)

(ii)

(iii)

(iv)

Power is given by, $P=F V$ $=\operatorname{mav}=\operatorname{mv} \frac{d v}{d t}$ vdv $=\frac{k}{m} \mathrm{dt}$ On integration, we get, 10 $\frac{v^{2}}{2}=\frac{k}{m} d t$...

### A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time is proportional to

(i)

(ii)

(iii)

(iv)

Let the body mass be $m$ Let the Acceleration be $a$ According to Newton's second law of motion, we have, $F=m a($ constant $)$ As, $\mathrm{a}=\frac{d v}{d t}=$ constant $d v=d t \times$ constant...

### Answer carefully, with reasons :

(a) In an inelastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

Is the total linear momentum conserved during the short time of an inelastic collision of two balls?

(b) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during a collision, not gravitational potential energy).

(a) In an inelastic collision, there will be a loss of kinetic energy. After a collision, the K.E is always lower than the K.E before the impact. In an inelastic collision, the system's total linear...

### Answer carefully, with reasons :

(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?

(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?

(a) In an elastic collision, the initial and ultimate kinetic energy are equal. There is no kinetic energy conservation when the two balls contact. It is transformed into kinetic energy. (b) In an...

### State if each of the following statements is true or false. Give reasons for your answer

(a) Work done in the motion of a body over a closed loop is zero for every force in nature.

(b) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.

(a) False. The work done by the conservative force on the moving body in a closed loop is zero. (b) True

### State if each of the following statements is true or false. Give reasons for your answer

(a) In an elastic collision of two bodies, the momentum and energy of each body is conserved.

(b) The total energy of a system is always conserved, no matter what internal and external forces on the body are present.

(a) False Both bodies' momentum and energy are conserved collectively, rather than separately. (b) False. External forces on the system have the ability to work on the body and modify the system's...

### Underline the correct alternative:

(a) The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system

(b) In an inelastic collision of two bodies, the quantities which do not change after the collision is the total kinetic energy/total linear momentum/total energy of the system of two bodies.

(a) External force Internal forces, regardless of their direction, cannot cause a change in momentum. As a result, the change in total momentum is proportional to the system's external force. (b)...

### Underline the correct alternative:

(a) When a conservative force does positive work on a body, the potential energy of the body increases/decreases/remains unaltered.

(b) Work done by a body against friction always results in a loss of its kinetic/potential energy.

(a) Decreases When a body is displaced in the direction of the force, the conservative force does positive work on it, causing the body to migrate to the center of force. As a result, the distance...

### Answer the following:

(a) An artificial satellite orbiting the earth in a very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth?(b) In the Figure, the man walks carrying a mass of on his hands. In Fig., he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of hangs at its other end. In which case is the work done greater?

Solution: (a) As the satellite approaches the Earth, its potential energy drops, and since the system's total energy should remain constant, the kinetic energy increases. As a result, the...

### Answer the following:

(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?

(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?

(a) The mass of the rocket is lowered when the shell burns up owing to friction. As per the law of conservation of energy, we have, Total energy $=$ kinetic energy $+$ potential energy $=m g...

### The potential energy function for a particle executing linear simple harmonic motion is given by , where is the force constant of the oscillator. For , the graph of versus is shown in Figure. Show that a particle of total energy moving under this potential must ‘turn back’ when it reaches .

Solution: Energy of the particle will be, $\mathrm{E}=1 \mathrm{~J}$ $\mathrm{K}=0.5 \mathrm{~N} \mathrm{~m}^{-1}$ $\mathrm{K} . \mathrm{E}=\frac{1}{2} \mathrm{mv}^{2}$ Based on law of conservation...

### Given in Figure, are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

(a) (b) The total energy is given by the relation, $E=K . E .+P . E$. So, $K_{. E}=E-P . E .$ There can never be a negative amount of kinetic energy. In the region where K.E. becomes negative, the...

### Given in Figure, are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

The total energy is given by the relation, $E=K . E .+P . E$. So, $K_{. E}=E-P . E .$ There can never be a negative amount of kinetic energy. In the region where K.E. becomes negative, the particle...

### A body of mass initially at rest moves under the action of an applied horizontal force of on a table with the coefficient of kinetic friction Compute the

(a) work done by the net force on the body in ,

(b) change in kinetic energy of the body in .

Mass of the body is given as $2 \mathrm{~kg}$ Horizontal force applied is given as $7 \mathrm{~N}$ Coefficient of kinetic friction is given as $0.1$ Acceleration produced by the applied force can be...

### A body of mass initially at rest moves under the action of an applied horizontal force of on a table with the coefficient of kinetic friction Compute the

(a) work done by the applied force in ,

(b) work done by friction in

Mass of the body is given as $2 \mathrm{~kg}$ Horizontal force applied is given as $7 \mathrm{~N}$ Coefficient of kinetic friction is given as $0.1$ Acceleration produced by the applied force can be...

### The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

Work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

Work completed is negative. It is noticed that the bob's orientation and the air resistance acting on it are in different directions.

### The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

(a) work done by friction on a body sliding down an inclined plane,

(b) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity

(a) The direction of motion of the object is opposite the direction of the frictional force, as can be seen. As a result, the work completed is negative. (b) The frictional force acting on an object...

### The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.

(b) work done by the gravitational force in the above case

(a) Work done is positive. It is obvious that both the force and the displacement are in the same direction. (b) It should be observed that the object's displacement is upward, but the force of...