Work, Energy, and Power

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### 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...

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### 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, $(M+m) v$...

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### 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,...

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### 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...

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### 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,...

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### 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 $m_{2}=0.4 \mathrm{~kg}$...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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....

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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...

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### 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.

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### 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...

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### 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...

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