Work, Energy and Momentum

Transcription

1 Work, Energy and Moentu Work: When a body oves a distance d along straight line, while acted on by a constant force of agnitude F in the sae direction as the otion, the work done by the force is tered as W Fd Now if the force F akes an angle with the direction of the displaceent d then, the work done is F W F d FdCos Diension: [W] = [ML T - ]. Unit: The SI unit of work is N- or Joule. d Work is a scalar quantity and it can be both positive and negative. If the coponent of the force is in the sae direction of the displaceent, work is positive and otherwise negative. Work done by a varying force: Suppose a body of oves along a straight line. We divide the displaceent into short segents, naely, x, x, x etc and assue that an approxiate force F acts for x, F for x and so on, and then the work done is W = F x + F x + F x +. If the nuber of segent are very large and size of each is very sall, then the work done x W Fdx, where x and x are initial and final positions of the body. x Potential energy: This is the energy possessed by an object due to its state or position. If an object is raised fro the ground to height then it gets gravitational potential energy due its position. If a spring is stretched then its gain potential energy due to its state. Kinetic energy: The energy due to the otion of an object is known as kinetic energy. I f the ass of an object is and the velocity is v, then its kinetic energy would be k v Gravitational potential energy: When a gravitational force acts on a body undergoing a displaceent, the force does work on the body, and as we shall see that this work can be expressed in ters of initial and final position of the body. If a body of ass oves with vertically fro a height y to a height y and the downward gravitational force on the body is its weight W, then the work done, in this case is represented as W y y ) g( y ) W g ( y Thus W g can be deterined fro the values of the quantity gy at the beginning and end of the displaceent. This quantity, the product of the weight and the height above origin of the coordinates, is called the gravitational potential energy U. U gy W gravitational g U U U PHY 0/Chapter-4

2 Elastic potential energy: Figure : Elastic potential energy Figure shows a body of ass on a level frictionless surface. One end of a spring is attached to the body and the other end of the spring is fixed. A force P is applied to stretch the spring. As soon as the slightest extension takes place, a force F is created within the spring which is opposite in direction to P. The force F is called an elastic force. If the force P reduces to zero, the elastic force restores the spring to its original unstretched condition. It ay therefore be referred to as a restoring force. In a displaceent fro a initial elongation x to a final elongation x the elastic restoring force does an aount of work given by W el kx kx The quantity /kx, one half the product of the force constant and the force constant and the square of the coordinate of the body, is called elastic potential energy of the body U. U kx Conservative and dissipative forces: In the case of gravitational potential energy and elastic potential energy, we have seen that, the work done is independent of the path followed by the body and depends only upon the initial and final position. In both case the total echanical energy reains constant and the force required for work done is a conservative force. On the other hand, the frictional force is path-dependent. In this case the longer the path between two given points, the greater the work. When the friction acts alone, the total echanical energy is not conserved. The friction is therefore called a non-conservative or dissipative force. The echanical energy of a body is conserved only when no dissipative forces act on it. Work energy theore: The work done on a body for a force is related very directly to the change in the body s otion that results. To develop the relationship we consider first a body of ass oving along a straight line under the action of a constant resultant force of agnitude F directed along the line. Suppose the speed increase fro v to v, while the body undergoes a displaceent s. Then we have v v as v v a s Hence the force F is F a Then the work done is W Fs W v v v s v v ( v s s v ) v PHY 0/Chapter-4 F P

3 The quantity /v, one-half the product of the ass of the body and the square of it Its velocity is called kinetic energy k, i.e. k v W k k k Thus the work done on a body is always equal to the change in the kinetic energy of the body. This stateent is known as the work energy theore. If the work is positive the final kinetic energy is greater than the initial kinetic energy and the kinetic energy increases. If the work done is negative, the kinetic energy decreases. In the special case in which the work is zero, the kinetic energy reains constant. Power: Tie considerations are not involved in the definition of work. It is iportant to consider the rate of the work. The rate at which work is done by a working agent is called the power developed by that agent. If a quantity of work W is done in a tie interval t then the average power is defined as Average power = Work done Tie interval W P t The instantaneous power is defined as W dw P Li t 0 t dt Diension: [P] = [ML T - ]. Unit: The SI unit of power is Js - or Watt. Power and velocity: Suppose a force F acts on a body while it undergoes a displaceent s along its path. Then the power can be obtained fro the equation W Fs P Fv t t In the liit t 0, P = Fv Law of conservation of energy: The law of conservation of energy state that, energy can neither be created nor can destroyed, it can only be changed fro one type to another and the total energy in the universe is constant. Moentu: The product of the ass and velocity of an object is called the oentu. It is also soeties called linear oentu and sybolized by p. Therefore, p = v. Unit: The SI unit of oentu is kgs -. Conservation of oentu: The principle of conservation of oentu is one of the ost fundaental and iportant principles of echanics. It states that, When no external forces acts on a syste, the total oentu of the syste reains constant in agnitude and direction. Let us suppose that two asses of and are oving toward each other with initial velocities v 0 and v 0. After collision, if the bodies ove with velocities v and v, then according to conservation of oentu v 0 + v 0 = v + v PHY 0/Chapter-4

4 Elastic collision: If the total energy before and after collision are the sae then this collision is called the elastic collision. Let us suppose that two asses of and are oving toward each other with initial velocities v 0 and v 0. After collision, if the bodies ove with velocities v and v, then K.E before collision, v v 0 K.E aftercollision, v Since the collision is elastic v0 v0 v v 0 v Center of ass: The center of ass of a syste of particles is a specific point at which, for any purposes, the syste's ass behaves as if it were concentrated. The center of ass is a function only of the positions and asses of the particles that coprise the syste. In the case of a rigid body, the position of its center of ass is fixed in relation to the object. In the case of a loose distribution of asses in free space, such as, say, shot fro a shotgun, the position of the center of ass is a point in space aong the that ay not correspond to the position of any individual ass. In the context of an entirely unifor gravitational field, the center of ass is often called the center of gravity the point where gravity can be said to act. Let us consider a collection of any nuber of particles whose total oentu P can defined as the vector su of the individual oentu, as P = p + p + p = v + v + v Now, if the velocity of the center of ass is V and the total ass of the body is M, then the total oentu can also be written as P MV v v v... v v v... V... This defines the velocity of the center of ass, and it also defines the position of the center of ass. Since the particle velocities are the tie derivatives of the position vectors r, r, r... of the particles, we define position vector of the center of ass, R as r r r... R... Exercise 6-0: A body of ass kg is initially at rest on a horizontal surface. If a horizontal force of 5N pushes it 4, using work energy relationship, find the velocity. The work-energy relationship is W v v We also know that W Fs Here, v = 0, F=5N, s = 4, =kg 5N 4 kg v kg v 0s PHY 0/Chapter-4 4

5 Exercise 6-: A body of ass 8kg oves in a straight line on horizontal surface. At one point in its path its speed is 4s -. After it has traveled its speed is 5s - in the sae direction. Use work-energy relationship and find the force acting on the body. Here, v 4s, v 5s, 8kg, s Fro the work-energy relationship W v v Fs v v v v F s 8kg 5s 8kg 4s F F N N N Exercise 6-4: What is the potential energy of an 800kg elevator at the top of a building 80 above street level? Assue in the street level the potential energy is zero. We know, the potential energy is U gy Here 800kg, y 80 U 6 800kg9.8 s J Exercise 6-50: The haer of a pile driver has a ass of 500kg and ust be lifted a vertical distance of in s. What horse power engine is required? We know, dw Fds gds P dt dt dt Here, 500kg, ds, dt s Now, kg9.8 s 500 P 66.67Js Watt s 746Watt H. P P H. P 4.8H. P. 746 Probles for practice: Exercise 7-, 7-(a) (b) (c), 7-(a) (b), 7-0(a) (b), 7-, etc. PHY 0/Chapter-4 5