Work, Power & Energy Notes

WORK, POWER & ENERGY Page # 1 WORK, POWER & ENERGY 1. WORK : Work is said to be done by a force when the force produces a displacement in the body on which it acts in any direction except perpendicular to the direction of the force. 1.1 Work done by constant force Consider an object undergoes a displacement S along a straight line while acted on a force F that makes an angle  with S as shown. The work done W by the agent is the product of the component of force in the direction of displacement and the magnitude of displacement. i.e., W = FS cos  ...(1) F S Work done is a scalar quantity and its S.I. unit is N-m or joule (J). We can also write work done as a scalar product of force and displacement.   ...(2) W  F. S where S is the displacement of the point of application of the force From this definition, we conclude the following points (A) work done by a force is zero if displacement is perpendicular to the force ( = 90°)  T  S Example. The tension in the string of a simple pendulum is always perpendicular to displacement. (Figure) So, work done by the tension is zero. (B) if the angle between force and displacement is acute ( < 90°), we say that the work done by the force is positive. Example : When a load is lifted, the lifting force and the displacement act in the same direction. So, work done by the lifting force is positive. Example : When a spring is stretched, both the stretching force and the displacement act in the same direction. So work done by the stetching force is positive. (C) If the angle between force and displacement is obtuse ( > 90°), we say that the work done by the force is negative. Example : When a body is lifted, the work done by the gravitational force is negative. This is because the gravitational force acts vertically downwards while the displacement is in the vertically upwards direction. __________________________________________________________________________________ Important points about work : 1. Work is said to be done by a force when its point of application moves by some distance.Force does no work if point of application of force does not move (S = 0) Example : A person carrying a load on his head and standing at a given place does no work. 2. Work is defined for an interval or displacement. There is no term like instantaneous work similar to instantaneous velocity. WORK, POWER & ENERGY Page # 2 =0.2 2kg frictionless 2kg 10 N 2m 10 N 2m Work done by 10 N force in both the cases are same = 20 N For a particular displacement, work done by a force is independent of type of motion i.e. whether it moves with constant velocity, constant acceleration or retardation etc. If a body is in dynamic equilibrium under the action of certain forces, then total work done on the body is zero but work done by individual forces may not be zero. When several forces act, work done by a force for a particular displacement is independent of other forces. A force is independent of reference frame. Its displacement depends on frame so work done by a force is frame dependent therefore work done by a force can be different in different reference frame. 3. 4. 5. 6. ________________________________________________________________________________________ 2. UNITS OF WORK : In cgs system, the unit of work is erg. One erg of work is said to be done when a force of one dyne displaces a body through one centimetre in its own direction.  1 erg = 1 dyne × 1 cm = 1 g cm s–2 × 1 cm = 1 g cm2 s–2 Note : Another name for joule is newton metre. Relation between joule and erg 1 joule = 1 newton × 1 metre 1 joule = 105 dyne × 102 cm = 107 dyne cm 1 joule = 107 erg 1 erg = 10–7 joule Dimensions of Work : [Work] = [Force] [Distance] = [MLT–2] [L] = [ML2T–2] Work has one dimension in mass, two dimensions in length and ‘–2’ dimensions in time, On the basis of dimensional formula, the unit of work is kg m2 s–2. Note that 1 kg m2 s–2 = (1 kg m s–2) m = 1 N m = 1 J. 3. WORK DONE BY MULTIPLE FORCES :     F If several forces act on a particle, then we can replace F in equation W = F . S by the net force where     F  F1  F2  F3 ......   F . S  W= ...(i)  This gives the work done by the net force during a displacement S of the particle. We can rewrite equation (i) as :       W = F1. S  F2 . S  F3 . S..... or W = W1 + W2 + W3 + ......... So, the work done on the particle is the sum of the individual work done by all the forces acting on the particle. A block of mass m is placed on an inclined plane which is moving with constant velocity v in horizontal direction as shown in figure. Then find out work done by the friction in time t if the block is at rest with respect to the incline plane.  Ex.1    v=const. m   WORK, POWER & ENERGY Sol. Page # 3 F.B.D of block with respect to ground. f f m v  v   m mg g sin  B  A mg mgsin Block is at rest with respect to wedge  f = mg sin  In time t the displacement of block with respect to ground d = vt Work done by friction for man A Wf = (component of friction force along displacement) × displacement Wf = mgsin.vt cos(180°–) Wf = – mg vt cos sin  Wf for man B = 0 (displacement is zero with respect to man B) 4. WORK DONE BY A VARIABLE FORCE : When F as a function of x, y, z When the magnitude and direction of a force vary in three dimensions, it can be expressed as a function of the position. For a variable force work is calculated for infinitely small displacement and for this displacement force is assumed to be constant.   dW = F. ds The total work done will be sum of infinitely small work B WA  B =    F. ds = A B    (F cos )ds A It terms of rectangular components,  F  Fx i  Fy j  Fzk  ds  dx i  dy j  dzk xB WA  B = 5.  xA yB Fx dx +  zB Fy dy + yA  F dz z zA AREA UNDER FORCE DISPLACEMENT CURVE : Graphically area under the force-displacement is the work done +ve work Fx Fy +ve work +ve work Fz y x –ve work z The work done can be positive or negative as per the area above the x-axis or below the x-axis respectively. 6. INTERNAL WORK : Suppose that a man sets himself in motion backward by pushing against a wall. The forces acting on the man are his weight 'W' the upward force N exerted by the ground and the horizontal force N exerted by the wall. The works of 'W' and of N are zero because they are perpendicular to the motion. The force N' is the unbalanced horizontal force that imparts to the system a horizontal acceleration. The work of N', however, is zero because there is no motion of its point of application. We are therefore confronted with a curious situation in which a force is responsible for acceleration, but its work, being zero, is not equal to the increase in kinetic energy of the system. WORK, POWER & ENERGY Page # 4 N' N The new feature in this situation is that the man is a composite system with several parts that can move in relation to each other and thus can do work on each other, even in the absence of any interaction with externally applied forces. Such work is called internal work. Although internal forces play no role in acceleration of the composite system, their points of application can move so that work is done; thus the man's kinetic energy can change even though the external forces do no work. "Basic concept of work lies in following lines Draw the force at proper point where it acts that give proper importance of the point of application of force. Think independently for displacement of point of application of force, Instead of relation the displacement of applicant point with force relate it with the observer or reference frame in which work is calculated.  displacement vector of po int of  W = (Force vector )   application of force as seen by  observer  7.      CONSERVATIVE FORCE : A force is said to be conservative if work done by or against the force in moving a body depends only on the initial and final positions of the body and does not depend on the nature of path followed between the initial and final positions. m m m m m m (a) (b) (c) Consider a body of mass m being raised to a height h vertically upwards as shown in above figure. The work done is mgh. Suppose we take the body along the path as in (b). The work done during horizontal motion is zero. Adding up the works done in the two vertical parts of the paths, we get the result mgh once again. Any arbitrary path like the one shown in (c) can be broken into elementary horizontal and vertical portions. Work done along the horizontal path is zero. The work done along the vertical parts add up to mgh. Thus we conclude that the work done in raising a body against gravity is independent of the path taken. It only depends upon the intial and final positions of the body. We conclude from this discussion that the force of gravity is a conservative force. Examples of Conservative forces. (i) Gravitational force, not only due to Earth due in its general form as given by the universal law of gravitation, is a conservative force. (ii) Elastic force in a stretched or compressed spring is a conservative force. (iii) Electrostatic force between two electric charges is a conservative force. (iv) Magnetic force between two magnetic poles is a conservative force. Forces acting along the line joining the centres of two bodies are called central forces. Gravitational force and Electrosatic forces are two important examples of central forces. Central forces are conservative forces. Properties of Conservative forces • Work done by or against a conservative force depends only on the initial and final position of the body. • Work done by or against a conservative force does not depend upon the nature of the path between initial and final position of the body. • Work done by or against a conservative force in a round trip is zero. If a body moves under the action of a force that does no total work during any round trip, then the WORK, POWER & ENERGY Page # 5 force is conservative; otherwise it is non-conservative. The concept of potential energy exists only in the case of conservative forces. • The work done by a conservative force is completely recoverable. Complete recoverability is an important aspect of the work done by a conservative force. Work done by conservative forces Ist format : (When constant force is given) 8. NON-CONSERVATIVE FORCES : A force is said to be non-conservative if work done by or against the force in moving a body depends upon the path between the initial and final positions. The frictional forces are non-conservative forces. This is because the work done against friction depends on the length of the path along which a body is moved. It does not depend only on the initial and final positions. Note that the work done by fricitional force in a round trip is not zero. The velocity-dependent forces such as air resistance, viscous force, magnetic force etc., are non conservative forces. 9. CONSERVATIVE FORCE AND POTENTIAL ENERGY : U s i.e. the projection of the force field , the vector F, at a given point in the direction of the displacement r equals the derivative of the potential energy U with respect to a given direction, taken with the opposite sign. The designation of a partial derivative /s emphasizes the fact of deriving with respect to a definite direction. So, having reversed the sign of the partial derivatives of the function U with respect to x, y, z, we obtain the projection Fx, Fy and Fz of the vector F on the unit vectors i, j and k. Hence, one can readily find the vector itself : Fs = –  U U U  F = Fxi + Fy j + Fzk, or F = –   x i   y j   z k    The quantity in parentheses is referred to as the scalar gradient of the function U and is denoted by grad U or  U. We shall use the second, more convenient, designation where  (“nabla”) signifies the symbolic vector or operator     = i x  j y  k  z Potential Energy curve : • A graph plotted between the PE of a particle and its displacement from the centre of force field is called PE curve. • Using graph, we can predict the rate of motion of a particle at various positions. dU • Force on the particle is F(x) = – dx Q S U B P1 A P2 R P O Case : I Case : II D C x On increasing x, if U increases, force is in (–) ve x direction i.e. attraction force. On increasing x, if U decreases, force is in (+) ve x-direction i.e. repulsion force. Different positions of a particle : Position of equilibrium If net force acting on a body is zero, it is said to be in equilibrium. For equilibrium R and S are the states of equilbrium positions. dU = 0. Points P, Q,, dx WORK, POWER & ENERGY Page # 6 Types of equilirbium : • Stable equilibrium : When a particle is displaced slightly from a position and a force acting on it brings it back to the initial position, it is said to be in stable equilibrium position. d2U dU = 0,and 2 =+ ve dx dx In figure P and R point shows stable equilibrium point. • Unstable Equilibrium : When a particle is displaced slightly from a position and force acting on it tries to displace the particle further away from the equilibrium position, it is said to be in unstable equilibrium. Necessary conditions:– d2U dU  0 potential energy is maximum i.e. = = – ve dx dx2 Q point in figure shows unstable equilibrium point • Neutral equilibrium : In the neutral equilibrium potential energy is constant. When a particle is displaced from its position it does not experience any force acting on it and continues to be in equilibrium in the displaced position. This is said to be neutral equilibrium. In figure S is the neutral point Condition : Condition : 10. d2U dU 0 , =0 dx dx2 WORK ENERGY THEOREM : If the resultant or net force acting on a body is Fnet then Newton's second law states that Fnet = ma ...(1) If the resultant force varies with x, the acceleration and speed also depend on x. a= v then dv dx ...(2) from eq. (1) Fnet dv = mv dx  Fnet.dx = m v dv m vi Fnet m vf vf F net . dx =  mvdv vi 1 1 mv 2f – mv i2 2 2 = kf – ki = K ...(3) Wnet = Wnet Wnet Work done by net force Fnet in displacing a particle equals to the change in kinetic energy of the particle i.e. we can write eq. (3) in following way (W.D)c + (W.D)N.C + (W.D)ext. + (W.D)pseudo = K where ...(4) (W.D)c = work done by conservative force (W.D)N.C = work done by non conservative force. (W.D)ext = work done by external force (W.D)pseudo = work done by pseudo force. we know that (W.D)c = – U WORK, POWER & ENERGY Page # 7  – U + (W.D)N.C + (W.D)ext + (W.D)pseudo = K  (W.D)N.C + (W.D)ext. + (W.D)pseudo = (kf + uf) – (ki + ui)  k + u = Mechanical energy.  work done by forces (except conservative forces) = change is mechanical energy. If (W.D)N.C = (W.D)ext = (W.D)pseudo = 0 Kf + Uf = Ki + Ui Initial mechanical energy = final mechanical energy This is called mechanical energy conservation law. Questions Based on work Energy Theorem : (A) When only one conservative force is acting 11. POWER Power is defined as the time rate of doing work. When the time taken to complete a given amount of work is important, we measure the power of the agent doing work. The average power (P or Pav ) delivered by an agent is given by W P or Pav =  t = Total work done Total time where W is the amount of work done in time t. Power is the ratio of two scalars-work and time. So, power is a scalar quantity. If time taken to complete a given amount of work is more, then power is less. • The instantaneous power is, P = • P= dW  where dW is the work done by a force F in a small time dt. dt  dr   dW   F. v where v is the velocity of the body.. = F. dt dt By definition of dot product, P = Fvcos   where  is the smaller angle between F and v This P is called as instantaneous power if dt is very small. 11.1 Unit of Power : A unit power is the power of an agent which does unit work in unit time. The power of an agent is said to be one watt if it does one joule of work in one second. 1 watt = 1 joule/second = 107 erg/second Also, 1 watt = 1 newton  1 metre = 1 N ms–1. 1 sec ond Dimensional formula of power [Power ]  [ Work ] [ML2 T –2 ] = = [ML2T–3] [ Time] [ T]