XI PHYSICS. M. Affan Khan LECTURER PHYSICS, AKHSS, K.

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1 XI PHYSICS M. Affan Khan LECTURER PHYSICS, AKHSS, K affan_414@live.com [WORK, POWER AND ENERGY] CHAPTER NO. 7 A little concept of vector mathematics is applied here and in which a little mistake may destroy the concept of gravitational mechanics. Also, Energy is a conserved quantity is proved in this chapter. Due to which we can derive Work-Energy Equation which is a very important tool for solving work and energy problems.

2 Work Work is defined as the product of displacement and the component of force in the direction of displacement. Mathematical form, Consider an object of mass m has to move displacement s when the force F at an angle θ is applied. Now the component which is responsible for its movement along the direction of displacement is F. W = F S W = (FcosθS W = FScosθ W = F. S Types of Work 1 Positive and Maximum work If the force applied is in the direction of displacement, then the angle between both vectors will be 0 0. Then, W = FScos0 0 W = FS(1 W = FS Zero Work If the force applied is perpendicular to the desired displacement, then the angle between both vectors will be W = FScos90 0 W = FS(0 W = FS 3 Negative Work In case of friction the body moves forwards and friction force acts in opposite direction tries to stop the body then the angle is between force and displacement, so that W = FScos180 0 W = FScos( 1 W = FS Units of Work: 1 Joule = Newton meter is S.I unit of work Erg = Dyne centimeter is C.G.S. unit of work 3 Foot Pound(lb is F.P.S unit of work

3 Conservative Force / Field A force for which work done around a closed path is zero or is independent of path between two fixed points, is called conservative force and its field is called as conservative field. E.g. Gravitational field, Electrostatic field etc. Non-conservative field A force for which work done around a closed path is not zero or is not independent of path between two fixed points, is called non-conservative force and its field is called nonconservative field. E.g. force of friction, magnetic force, viscous force Gravitational Force / Field is conservative To prove that gravitational force/field is conservative we have to consider a closed path. For this reason, we are considering a closed path in the form of triangle as shown in figure. If the work done around ABCA is zero, then the force is called conservative force and the field is called conservative field. Therefore, work done can be written as, W TOTAL = W A B + W B C + W C A Work done from A to B: W A B = F. S 1 = FS 1 cos(180 β where, (180 β is the angle between force and displacement W A B = FS 1 ( cosβ W A B = mgs 1 cosβ W A B = mgh Work done from B to C: W B C = F. S = FS cosα = mgs cosα where, α is the angle between force and displacement = mgh (S cosα = h Work done from C to A: W C A = F. S 3 = FS 3 cos90 0 W C A = mgs 3 (0 W C A = 0 Now the total work done can be written as, W TOTAL = W A B + W B C + W C A W TOTAL = mgh + mgh + 0 W TOTAL = 0 A S 1 β S 3 B α h S C

4 Energy: Forms of Energy 1 Mechanical Energy Chemical Energy 3 Electrical Energy 4 Heat Energy 5 Solar Energy 6 Nuclear Energy and others Types of Mechanical Energy: a Kinetic Energy b Potential Energy Since the total work done around this closed path is zero its mean gravitational force is a conservative force. Capability of doing work. Kinetic Energy: The energy possessed by a body due to its motion called as Kinetic Energy. Mathematical Expression, Consider a body of mass m is subjected to an external force F so that the body covers some displacement S as its velocity from zero to becomes v and its acceleration is a. Since it is in motion therefore the work W is done in the form of kinetic energy K.E. we may write, as = v f v i as = v 0 S = v a And work is given as, W = F. S W = Fscos0 0 W = (ma v a W = m(v W = 1 mv v i = 0 v f = v

5 The work stored in the form of Kinetic Energy W = K. E Therefore, K. E = 1 mv Work-Energy Equation: Work is also defined as change of energy. Consider an object of mass m moving with initial velocity v i subjected to some external force F which provides it some acceleration a to change its velocity to v f. In this time t the object is displaced S units. So, we may write as = v f v i S = v f v i a Then work done is given as, W = F. S W = Fscos0 0 W = (ma v f v i a W = m(v f v i W = 1 mv f 1 mv i W = (K. E f (K. E i W = K. E v i v f Potential Energy: The energy stored in a body due to its position is called as Potential Energy. Types of Potential Energy: 1. Gravitational Potential Energy. Elastic Potential Energy 3. Electrostatic Potential Energy and many others. Gravitational Potential Energy: The energy possessed by a body due to its height is called as gravitational potential energy. P. E = mgh

6 Explanation: Suppose an object of mass m is subjected to an upward force F which is applied to raise the object to a height h from the surface of Earth at a very slow rate. The slow movement is only possible when the applied force is equal to weight (mg of the body. We have work done here as, W = F. S W = FScosθ W = (mg(hcos0 0 W = mgh Here the work done is stored in the form of potential energy, P. E = mgh Power The rate of doing work is power. OR Energy consumed per unit time is called as power. Instantaneous Power: P av = W t W P ins = lim t 0 t Units Its S.I unit is Watt (i.e. J/s Also, can be written with following prefixes 1 MW = 10 6 Watts 1 GW = 10 9 Watts It has also unit of horse power. 1 hp = 746 Watts Another unit is foot-pound/second. 1 ft lb/s = 1.35 Watts Since we know that Therefore, Or in terms units, Power = Work Time Work = Power Time Work = Watt second Work = 1kwh = J

7 Power in terms of Force & Velocity: As we know that Power = Work Time P = W t P = F. S t P = FScosθ t P = Fcosθ ( S t P = Fcosθ(v P = Fvcosθ P = F. V Therefore, power is also defined as dot product of Force and Velocity. Law of Conservation of Energy Statement Energy can neither be created nor be destroyed but it can be changed from one form to another form, therefore, the total energy of the system remains constant. Explanation Consider an object with mass m situated at a certain height h from the ground level as shown in figure, if it is allowed to drop from this height then it will fall freely under the action of gravity. A At Point A Since at this point the object is at its maximum height h and with x B initial velocity zero, therefore it has, P. E = mgh h K. E = 0 E = K. E + P. E C E = 0 + mgh E = mgh At Point B When it will reach at point B, it has covered displacement x from its initial position and we can say according to diagram that it is now located at height h-x. Also, let us call velocity of this object at this point as v B. P. E = mg(h x

8 K. E = 1 mv B For v B as = v f v i gx = v B 0 v B = gx K. E = 1 m(gx K. E = mgx Now, the total energy is E = K. E + P. E E = mgx + mg(h x E = mgx + mgh mgx E = mgh At Point C When it will reach ground level, the height from surface of the Earth will be zero, and let us call this velocity as v c. P. E = mg(0 = 0 K. E = 1 mv c For v c as = v f v i gh = v C 0 v c = gh K. E = 1 m(gh K. E = mgh Now, the total energy at this point will be, E = K. E + P. E E = mgh + 0 E = mgh Hence it is proved that Energy converts from one form to another form and remains conserved. We can conclude for this case that loss of potential energy = gain of kinetic energy If there was air friction present, then the object which was coming downwards had to perform work done against friction also, loss of potential energy work done against friction = gain of kinetic energy

9 loss of potential energy = gain of kinetic energy + work done against friction mgh = 1 mv + fh There are many other cases where we can define Energy Conservation as we did above, E.g. 1 As we rub our hands it produces heat, then we may write, mechanical energy = heat energy + losses As fuel burns to run engine then, chemical Energy = mechanical energy + losses Absolute Gravitational Potential Energy We measure gravitational potential energy from the surface of Earth (i.e. mgh, which states that at surface of the Earth h = 0 then P. E = 0, which should not be the case since the phenomenon of attraction take place from the center of mass, therefore, at surface of Earth P.E cannot be equal to zero. That s why to find the absolute gravitational potential energy let us perform a hypothetical experiment. Consider an object of mass m is placed at certain distance from the center of the Earth, we ll be performing work to raise this object from this place to infinity (in this case out of the gravitational field of Earth. Let us now consider a force F is applied on it to raise it to required height. For simplicity, we can divide this displacement in n intervals given that for all intervals the force is approximately same. As we move from position 1 to, the displacement covered is, r = r r 1 And since force is approximately same on each point since displacement is very small therefore, we can take average of these two forces F = F 1 + F F = 1 (GM Em + GM Em r 1 r F = GM Em ( 1 r r F = GM Em ( r + r 1 r 1 r From diagram, we may write, r = r + r 1

10 F = GM Em ( ( r + r 1 + r 1 r 1 r F = GM Em ( ( r + rr 1 + r 1 + r 1 r 1 r Since r is very small, therefore we can neglect its square term. F = GM Em ( rr 1 + r 1 r 1 r F = GM Em (r 1 ( r + r 1 r 1 r F = GM E m ( r + r 1 r 1 r F = GM E m r r 1 r F = GM Em r 1 r Therefore, work done from 1 st to nd position can be given as, W 1 = F. r W 1 = F rcosθ W 1 = F rcos180 0 W 1 = F r W 1 = GM Em r 1 r (r r 1 W 1 = GM E m r r 1 r 1 r W 1 = GM E m ( r r 1 r r 1 r 1 r W 1 = GM E m ( 1 r 1 1 r This is the work done when object was displaced from r 1 to r. Similarly work at other positions will be given as, W 3 = GM E m ( 1 r 1 r 3 W 3 4 = GM E m ( 1 r 3 1 r 4... W n 1 n = GM E m ( 1 r n 1 1 r n

11 Let us find total work, W TOTAL = W 1 W 3 W 3 4 W n 1 n W TOTAL = ( GM E m ( 1 r 1 1 r + ( GM E m ( 1 r 1 r 3 + ( GM E m ( 1 r 3 1 r ( GM E m ( 1 r n 1 1 r n W TOTAL = GM E m ( 1 r 1 1 r + 1 r 1 r r r r n 1 1 r n W TOTAL = GM E m ( 1 r 1 1 r n This work done is stored in the form of Potential Energy P. E = GM E m ( 1 r 1 1 r n Since r n is a point which is outside the Earth s gravitational field, therefore, for absolute Potential Energy, we may take r n = (P. E abs = GM E m ( 1 r 1 1 (P. E abs = GM E m ( 1 r 1 0 (P. E abs = GM Em r 1 For any point r, we may write, (P. E abs = GM Em r Since r is in denominator which shows that as we go away from the center of Earth, P.E will decrease and as r approaches, P.E becomes zero. Also, negative sign indicates the bound condition of objects around Earth that is all objects in the Earth s gravitational field. At surface of Earth, r = R E (P. E = GM Em R E At some height h above Earth s surface, then r = R E + h (P. E abs = GM Em R E + h (P. E abs = GM Em R E (1 + h R E (P. E abs = GM Em (1 + h 1 R E R E

12 Applying Binomial Theorem, (P. E abs = GM Em R E (1 h R E + h R E + Neglecting squares and higher powers we get, (P. E abs = GM Em R E (1 h R E