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WORK POWER & ENERGY WORK, POWER & ENERGY WORK Whenever a force is applied to an object, it is ready to do work but work will be done only if it displaces the object. In work always at least two bodies are involved. One who is doing work (whose energy is decreasing). The other on which work is being done (whose energy is increasing). Fig. (a) If a man punches a hard wall, the energy he wishes to transfer to wall (to do work on wall), is reflected back to his hand and he will be injured as no displacement takes place on wall, hence no work is done and obviously no energy is used by the wall. fig. (a) Sa | Ex &S Fig. (b) If the man makes his punch strong enough to break the wall, his hand will not be injured, the reason is the utilization of his energy by the wall (work done in breaking). As the wall breaks, displacement is produced by the force of his punch, hence energy is transferred to the wall and negligible amount of energy is reflected. 2 E fig. (b) The concept of work is now clear that it is just the transfer of energy due to the displacement produced by the applied force. How much work we do depends on both levels as to how hard we push and how far we move the object. In physics, the meaning of work is more precise and restricted. If we exert a constant force F on an object, causing it to move a distance parallel to F, then the work W done by the force is defined to be product of the magnitude of the force times the distance through which it acts as the object is moved. WORK DONE BY A CONSTANT FORCE 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. Work depends upon two factors : (i) Force must be applied. (ii) Distance travelled by the body in the direction of the force. Work done by the force is measured by the product of magnitude of force and the displacement of the point of application in the direction of force. i.e. W = FS. WORK POWER & ENERGY Let a force F moves through a displacement Ss and say the directions of these two vectors are not the same. If the angle between the displacement vector and the force vector is 0. The component of the force in the direction of the displacement is F cos 0. Work done = component of force in the direction of the displacement * magnitude of displacement. W = (Fcos0) S = F S cos 0 FE or W = Fs |—-> F cos 6 <————__§————+ Although work is the dot product of two vector quantities force and displacement, it is itself a scalar quantity. (a) By the definition of work the aspect which must be emphasized is that only the component of the force parallel to the displacement contributes to the work performed. (b) Any arbitrary force can always be resolved into two components, one parallel and one perpendicular to the displacement of the particle, and only force in the direction of the displacement contributes to the work. Examples of Zero Work — ey Case (1a) F, 9 When 6 = 90°, then W = FS cos 90° = 0 Even though the centripetal force mv?/r acts during a dis- placement, but no work is performed. The reason for this is that the force is perpendicular to the displacement. If a body sliding over a smooth horizontal surface. The work done by the force of gravity and the reaction of the surface will be zero. This is because both the force of gravity and the reaction act normally to the displacement. The same argument can be applied to a man carrying a load on his head and walking on a railway platform. The tension inthe string of asimple pendulum is always perpendicular to displacement. So, work done by the tension is zero, Case (1b) When S = 0, then W = 0. So, work done by a force is zero if the body suffers no displacement on the application of a force. A person carrying a load on his head & standing at a given place does no work. A man pushing against a wall would get tired but the mechanical work done would be zero, as there is no displacement. 2 WORK POWER & ENERGY Examples of Positive Work I When 0° =< 6 < 90°, then cosé is positive I Positive work means that force or its component along the displacement is also in the direction of the displacement. ‘Ar When a body is pulled on a fixed rough surface, the work done by the pulling force is positive as the pulling force is always in the direction F. of the displacement. I When a horse pulls a cart, the applied force and the displacement are ahs — in the same direction. So, work done by the horse is positive. ‘ ’ posit _ é When a load is lifted, the lifting force and the displacement act in the The earth¥s orev tational same direction. So, work done by the lifting force is positive. force does positive work When a spring is stretched, both the stretching force and the displacement on:a ball going down act in the same direction. So, work done is positive. ; | Examples of Negative Work | When 90° < 0 = 180°, then cos) is negative Work done by a force is said to be negative if the applied force has a component in a direction opposite to that of the displacement. j i ES The frictional force does negative work for both parts of a | round-trip. The work for the round-trip is not zero. | go, Initial ' } position Initial Final NL positi on position The earth's gravitational He oy ar ee: are force does negative work j : a i F F ona ball going up (a) (b) When brakes are applied to a moving vehicle, the work done by the braking force is negative. This is because the braking force and the displacement act in opposite directions. 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 upward direction. Work done can be positive, negative or zero. F ~~ + Food , ro = —*F ’ d ——> J positive work negative work zero work on gh ha 3 -) [o<0<5) [5 When the force varies linearly with the distance aways from some particular point. The example of this case is a spring. When a spring is either stretched or compressed by an amount from its normal position the force that acts to pull or push the spring back again is directly proportional to x. We can write for this force F = —-k x. Here, the minus sign indicates that the force is in the direction opposite to displacement x. The constant k is the spring constant. As the force of the spring resists any change in the length of the spring, we must apply a force directed oppositely to the force of the spring. Thus the force applied is in the same direction as the displacement x of the spring from its equilibrium position. The external force F = kx Work done by external force when spring is elongated from x, to x, Work done in small displacement dx is dW = Fdx x Rp Total work done We frax = k Jax Hy xy 1 1 W = 2 kx? -— > kx? Ex. Calculate work done from the graph Work done = area of triangle (OAB) — area of rectangle (BCDE) 1 We az * 20% 10- 10x 5 = 50 J WORK POWER & ENERGY CONSERVATIVE FORCE A force is said to be conservative if the work done by or against the conservative force : (a) Is independent of path and depends only on initial and final positions (b) Does not depend on the nature of path followed between the initial and final positions. Examples of Conservative Force All central forces are conservative like gravitational, electrostatic, elastic force, restoring farce due to spring etc. In presence of conservative forces mechanical energy remains constant. To summarise with [1] When force or more than one forces are acting on a particle in such a way that if particle come back to its initial position and kinetic energy of particle is equal to its previous value which was initially, than forces are known as conservative forces. [2] A force is conservative if the work done by it on a particle that moves between two points a and b along two differents paths A and B. Such that W, = W,. [3] If work done by a force acting on a particle over a close path is zero then force is conservative W,, + W,, = 0 or W,, = — W,.. These three defination are statements of the same thing in different way. Special Point : (a) Work done along a closed path or in a cyclic process is zero. i.e. f Fax-0 (b) If F conservative force then Fx F-0.- CENTRAL FORCES The force whose line of action always passes through a fixed point (which is known as centre of force) and magnitude of force depends only on the distance from this point is known as central force. + « rg ‘i F = F(t) (t) F = & FQ) [he rey All forces following inverse square law are central forces. ik S Fe= 12 r is central force like Gravitational force and Columb force. lf central force act on a particle then torque acting on the particle is t=1xP=rxF(r)i * > » * =p: uy aT HF Let eo (7 rxr=0) r r t=O 2a as dt dt i: J = constant t WORK POWER & ENERGY When a particle moves under the action of central force then its angular momentum remains conserved. 1. All central forces are conservative forces 2. Central forces are function of position 3. The angular momentum of an object moving under a central force is conserved NON CONSERVATIVE FORCE 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. Work done in a closed path is not zero in a non-conservative force field. 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 frictional force in a round trip is not zero. Examples of non-conservative force The velocity-dependent forces such as air resistance, viscous force etc. are non-conservative forces. S.No.| Conservative Forces Non-conservative Forces 1, Work done does not depend upon path. Work done depends upon path. os Work done in a round trip is zero. Work done in a round trip is not zero. 3. Central forces, spring forces etc. Force are velocity-dependent & retarding in nature, 4. When only a conservative force acts Work done against a non-conservative force within a system, the kinetic energy may be dissipated as heat energy. and potential energy can change. However, their sum, the mechanical energy of the system, doesnot change. 5, Work done is completely recoverable. Work done is not completely recoverable. 8 WORK POWER & ENERGY ENERGY The energy of a body is defined as the capacity of doing work or ability of the body to do work. It is a scalar quantity. The dimensional formula of energy is [ML*T~]. It is the same as that of work. The units of energy are the same as those of work i.e, joule and erg. Energy of a body give us an idea of the total amount of work that the body can do. It has nothing to do with the time taken to do the work. Energy is of many types - mechanical energy, sound energy, heat energy, light energy, chemical energy, atomic energy, nuclear energy etc. In many processes that occur in nature energy may be transformed from one form to other. Mass can also be transformed into energy and vice-versa. In dynamics, we are mainly concerned with purely mechanical energy. Law of Conservation of Energy The study of the various forms of energy and of transformation of one kind of energy into another has led to the statement of a very important principle, known as the law of conservation of energy. “Energy cannot be created or destroyed; it may only be transformed from one form into another. As such the total amount of energy never changes". WORK ENERGY THEOREM Consider a constant force F acting on a particle of mass m. Let a acceleration ‘a’ be produced in the direction of force F , Say along X-axis. Let the resultant force vary in magnitude only, not in direction. The work done by the resultant force in displacing the particle from x, to x is ft f w=[F _ ox -[F ax i i dv _ dv dx | dv i = a= = * = Since F = ma, and Ged Gh Vx ft ft r ft ry2 Hence, w=[ma dx = [mv &- dx =m [v-dvem(5) i i i 1 1 or W = Smvi ~5mvi "The work done by the resultant force acting on a particle is equal to the change in the kinetic energy of the particle”. 121s W= imi ~ > mvj Work Energy Theorem. Let a particle of mass m is moving with velocity v. This particle is stopped by some resistive force i.e. stopping force and due to action of this force particle comes in rest. Then W =0-—mv? = pv? WORK POWER & ENERGY Negative sign shows that particle is doing work against the stopping force. It means capacity of doing 1 work of a moving particle is 3m. So that it is known as kinetic energy of particle. K=—mv? The energy possessed by a body by virture of its motion is known as kinetic energy. The work done on a particle by the resultant force is equal to the change in Kinetic energy of the particle. So, the work done by the resultant force Ww =K,-K = AK Salient features of the work energy theorem & When the speed of the particle is constant, there is no change in Kinetic energy and the work done by the resultant force is zero. For example, in case of uniform circular motion, the speed of the particle is constant and so the centripetal force does no work on the particle. * It can also be concluded that if no external work is done, the Kinetic energy before a process must be equal to its Kinetic energy at the conclusion of the process. ve Kinetic energy of a particle decreases by an amount just equal to the amount of work which the particle performs. A body is said to have energy associated with it because of its motion; as it does work it slows down and loses some of this energy. % Work and energy are interchangeable quantities. When work is done, it appears as energy. The energy can be decreased by permitting the particle to do work on other particles. KINETIC ENERGY The KE of a moving body is equal to the amount of work that must be done to bring a body from rest into the state of motion. Conversely, the amount of work that we must do in order to bring a moving body to rest is equal to the negative of the kinetic energy of the body, i.e., KE work done to put the body into motion —work done to bring the body to a stop As mass m and v2(y . v) are always positive, kinetic energy is always positive scalar, i.e., kinetic energy can never be negative. KE is always positive. The KE is a scalar quantity. (a) Suppose a block is at rest on a frictionless surface and a constant net force F acts on the block. If v be velocity acquired by the block after travelling a distance x, then KE is 1 K = W = Fx = max = 5 mv? [w v? = 2ax] 10 WORK POWER & ENERGY (b) If the block is already in motion when the constant force is applied to it, then the work done is equal to change in KE of the block, i.e. (c) KE depends on the frame of reference, e.g., KE of a person of mass m sitting in a train moving with speed v is zero w.r.t. frame of train but (1/2) mv? w.rt. frame of reference of earth. Ex. Ina ballistics demonstration, a police officer fires a bullet of mass 50.0 g with speed 200 ms” on soft plywood of thickness 2.00 cm. The bullet emerges with only 10% of its initial kinetic energy. What is the emergent speed of the bullet? 1 50 Sol. Initial kinetic energy, K, = =~ 200 x 200J = 1000) 2.4000" 10 Final kinetic energy, K; = Foo * 12004 = 100J If v, is emergent speed of the bullet, then xvi =100 of v? = 4000 or v, = 63.2 ms”. Note that the speed is reduced by approximately 68% and not 90%. Ex. A body of mass 5 kg initially at rest is subjected to a force of 20 N. What is the kinetic energy acquired by the body at the end of 10 s ? 1 Sol. m = 5 kg; u = 0, F = 20N;t = 105, gm =? queer calling gO m 5 v=u+ at= 0+ 4 * 10 ms? = 40 ms’ , K.E. = 5 * 5 kg * (40 ms“’ = 4000 joule Ex. Determine the average force necessary to stop a bullet of mass 20 g and speed 250 ms” as it penetrates wood to a distance of 12 cm. Sol. If F newton be the retarding force, then the work done by force is given by W=F =x S =F x 0.12 joule 1 Loss of kinetic energy = > 5° Toop * 250 * 250 joule = 625 joule (This kinetic energy is consumed in stopping the bullet and is converted into heat energy) Applying work-energy theorem, F x 0.12 = 625 _ 625 2 N=5. 2N 012 5.2% 10 or F It is interesting to note that the retarding force is nearly 30,000 times the weight of the bullet. 11 WORK POWER & ENERGY Q. What is the stopping distance for a vehicle of mass m moving with speed v along a level road, if the coefficient of friction between the tyres and the road is 1? Ans. When the vehicle of mass m is moving with velocity v, the kinetic energy of the vehicle K = (1/ 2)mv* and if s is the stopping distance, work done by friction W = fs cos ® =pmgscosi80=-umgs. So by Work-Energy Theorem, W = AK = K, - K, . 41 ie., —umgs=d0 ~ zm? or s= 35. Special Points : 1. If kinetic energy increases (K, > K) then work done is positive. 2. If kinetic energy decreases (K, < K) then work done is negative. 3. If kinetic energy remains unchanged (K, = K) then work done is zero. e.g. circular motion POTENTIAL ENERGY The energy stored in a body or system by virute of its configuration or its position in a field is called potential energy. The potential energy is equal to negative of work done in shifting an object from some refernece position to a given position for conservative forces. + + = + there for au=-[F .dr or U,-U;=—-] F. dr 1. Regarding potential energy U it is worth noting that : (a) Potential energy can be defined only for conservative forces. It does not exist for non-conservative forces. (b) Potential energy can be positive, negative or zero. (c) Potential energy depends on frame of reference. (d) A moving body may or may not have potential energy. (e) Potential energy should be considered to be a property of the entire system, rather than assigning it to any specific particle. 2. Potential energy depends on position of reference level. At reference level F = 0 and potential energy U = 0 ki e.g. (a) for gravitational force F=-oMm (b) for electrostatic force F = — r F=Qatr=m so we select reference level at 2 Pp > > and hence potential energy at point P is u--[F .dr=—-W 12 WORK POWER & ENERGY (c) for springs F = — kx F = 0 at x = 0 so reference level is natural length of spring. * + + * = + U= JF dx =—[—kx.dx a Unt 0 0 2 (d) for intermolecular forces reference level is at infinity. 3. Potential energy depends on nature of force (a) for attractive forces U is negative (b) for repulsive forces U is positive 4, When work is done by the force i.e. body moves in direction of force potential energy will decrease. a When work is done against the force i.e. body is displaced opposite to direction of force potential energy will increase. 6. For potential energy determination we do not include the work done against non-conservative forces. smooth a \ jane plane “Y ;, Pp h work done = mgh + work done against friction potential energy = work done = imgh but potential energy = mgh Law of Conservation of Mechanical Energy The sum of the potential energy and the kinetic energy is called the total mechanical energy. Total mechanical energy of a system remains constant if only conservative forces are acting on a system of particles and the work done by all other forces is zero. AK + AU = 0 ao K-K+U,-U,=0 or K, + U, = K, + U, = constant 13 WORK POWER & ENERGY Examples based on Law of conservation of mechanical energy. i (a) Freely falling body : At the highest point, total eneray is in the form $4 : | u=0 of potential energy. At an intermediate point, energy is in the form of both PE and KE. At the lowest point, total energy is in the form of only KE. (KE), h | i ily TE (PE), = (KE),, + (PE),, 1 mgh = mgh, + > mv? TE 3 Ul | 3 <= uy r <= (b) Body projected vertically upwards : At the starting point total energy is in the form of KE only, at an intermediate point energy is in the form of both PE and KE and at the highest point total energy is in the form of only PE. 
