Work, Power and Energy || Class 11 Chapter 6 notes || Notes ||
The terms ‘work’, ‘energy’ and ‘power’ are frequently used in everyday language. A farmer ploughing the field, a construction worker carrying bricks, a student studying for a competitive examination, an artist painting a beautiful landscape, all are said to be working. In physics, however, the word ‘Work’ covers a definite and precise meaning. Somebody who has the capacity to work for 14-16 hours a day is said to have a large stamina or energy. We admire a long distance runner for her stamina or energy. Energy is thus our capacity to do work. In Physics too, the term ‘energy’ is related to work in this sense, but as said above the term ‘work’ itself is defined much more precisely. The word ‘power’ is used in everyday life with different shades of meaning. In karate or boxing we talk of ‘powerful’ punches. These are delivered at a great speed. This shade of meaning is close to the meaning of the word ‘power’ used in physics. We shall find that there is at best a loose correlation between the physical definitions and the physiological pictures these terms generate in our minds. The aim of this chapter is to develop an understanding of these three physical quantities. Before we proceed to this task, we need to develop a mathematical prerequisite, namely the scalar product of two vectors.
We have learnt about vectors and their use in Chapter 4. Physical quantities like displacement, velocity, acceleration, force etc. are vectors. We have also learnt how vectors are added or subtracted. We now need to know how vectors are multiplied. There are two ways of multiplying vectors which we shall come across : one way known as the scalar product gives a scalar from two vectors and the other known as the vector product produces a new vector from two vectors. We shall look at the vector product in Chapter 7. Here we take up the scalar product of two vectors. The scalar product or dot product of any two vectors A and B, denoted as A.B (read A dot B) is defined as A.B = A B cos θ (6.1a) where θ is the angle between the two vectors as Since A, B and cos θ are scalars, the dot product of A and B is a scalar quantity. Each vector, A and B, has a direction but their scalar product does not have a direction. From Eq. (6.1a), we have A.B = A (B cos θ ) = B (A cos θ ) Geometrically, B cos θ is the projection of B . A and A cos θ is the projection of A onto B.So, A.B is the product of the magnitude of A and the component of B along A. Alternatively, it is the product of the magnitude of B and the component of A along B. Equation (6.1a) shows that the scalar product follows the commutative law : A.B = B.A Scalar product obeys the distributive law: A. (B + C) = A.B + A.C Further, A. (λ B) = λ (A.B) where λ is a real number. The proofs of the above equations are left to you as an exercise. For unit vectors i, j,k we have ii j j kk ⋅=⋅= ⋅ = 1 i j jk ki ⋅=⋅ = ⋅= 0 Given two vectors A i jk =++ AAA xyz B i jk =++ BBB xyz their scalar product is ( ) ( ) . ˆˆˆ ˆˆˆ . AB i j k i j k = ++ ++ AAA BBB xyz xyz = AB AB AB + + xx yy zz (6.1b) From the definition of scalar product and we have : ( i ) A A =++ AA AA AA xx yy zz . Or, A AAA 2 x 2 y 2 z 2 =++ (6.1c) since A.A = |A ||A| cos 0 = A2. (ii) A.B = 0, if A and B are perpendicular.
Work is related to force and the displacement over which it acts. Consider a constant force F acting on an object of mass m. The object undergoes a displacement d in the positive x-direction .
The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement. Thus W = (F cos θ )d = F.d
As noted earlier, if an object of mass m has velocity v, its kinetic energy K is 2 K m mv 1 1 = = 2 2 v v. (6.5) Kinetic energy is a scalar quantity. The kinetic energy of an object is a measure of the work an object can do by the virtue of its motion. This notion has been intuitively known for a long time. The kinetic energy of a fast flowing stream has been used to grind corn. Sailing ships employ the kinetic energy of the wind. Table 6.2 lists the kinetic energies for various objects.
The word potential suggests possibility or capacity for action. The term potential energy brings to one’s mind ‘stored’ energy. A stretched bow-string possesses potential energy. When it is released, the arrow flies off at a great speed. The earth’s crust is not uniform, but has discontinuities and dislocations that are called fault lines. These fault lines in the earth’s crust are like ‘compressed springs’. They possess a large amount of potential energy. An earthquake results when these fault lines readjust. Thus, potential energy is the ‘stored energy’ by virtue of the position or configuration of a body. The body left to itself releases this stored energy in the form of kinetic energy. Let us make our notion of potential energy more concrete. The gravitational force on a ball of mass m is mg . g may be treated as a constant near the earth surface. By ‘near’ we imply that the height h of the ball above the earth’s surface is very small compared to the earth’s radius RE so that we can ignore the variation of g near the earth’s surface*. In what follows we have taken the upward direction to be positive. Let us raise the ball up to a height h. The work done by the external agency against the gravitational force is mgh. This work gets stored as potential energy. Gravitational potential energy of an object, as a function of the height h, is denoted by V(h) and it is the negative of work done by the gravitational force in raising the object to that height. V (h) = mgh If h is taken as a variable, it is easily seen that the gravitational force F equals the negative of the derivative of V(h) with respect to h. Thus, d d F V(h) m g h =− =− The negative sign indicates that the gravitational force is downward. When released, the ball comes down with an increasing speed. Just before it hits the ground, its speed is given by the kinematic relation, v2 = 2gh This equation can be written as 2 1 m v2 = m g h which shows that the gravitational potential energy of the object at height h, when the object is released, manifests itself as kinetic energy of the object on reaching the ground.
6.Power
Often it is interesting to know not only the work done on an object, but also the rate at which this work is done. We say a person is physically fit if he not only climbs four floors of a building but climbs them fast. Power is defined as the time rate at which work is done or energy is transferred. The average power of a force is defined as the ratio of the work, W, to the total time t taken P W t av = The instantaneous power is defined as the limiting value of the average power as time interval approaches zero, d d W P t = (6.21) The work dW done by a force F for a displacement dr is dW = F.dr. The instantaneous power can also be expressed as d d P t = F. r = F.v (6.22) where v is the instantaneous velocity when the force is F. Power, like work and energy, is a scalar quantity. Its dimensions are [ML2T–3]. In the SI, its unit is called a watt (W). The watt is 1 J s–1. The unit of power is named after James Watt, one of the innovators of the steam engine in the eighteenth century. There is another unit of power, namely the horse-power (hp) 1 hp = 746 W This unit is still used to describe the output of automobiles, motorbikes, etc. We encounter the unit watt when we buy electrical goods such as bulbs, heaters and refrigerators. A 100 watt bulb which is on for 10 hours uses 1 kilowatt hour (kWh) of energy. 100 (watt) × 10 (hour) = 1000 watt hour =1 kilowatt hour (kWh) = 103 (W) × 3600 (s) = 3.6 × 106 J Our electricity bills carry the energy consumption in units of kWh. Note that kWh is a unit of energy and not of power.
In the previous section we have discussed mechanical energy. We have seen that it can be classified into two distinct categories : one based on motion, namely kinetic energy; the other on configuration (position), namely potential energy. Energy comes in many a forms which transform into one another in ways which may not often be clear to us.
7.1 Heat:
We have seen that the frictional force is not a conservative force. However, work is associated with the force of friction, Example 6.5. A block of mass m sliding on a rough horizontal surface with speed v0 comes to a halt over a distance x0. The work done by the force of kinetic friction f over x0 is –f x0. By the work-energy theorem 2 m v /2 f x . o = 0 If we confine our scope to mechanics, we would say that the kinetic energy of the block is ‘lost’ due to the frictional force. On examination of the block and the table we would detect a slight increase in their temperatures. The work done by friction is not ‘lost’, but is transferred as heat energy. This raises the internal energy of the block and the table. In winter, in order to feel warm, we generate heat by vigorously rubbing our palms together. We shall see later that the internal energy is associated with the ceaseless, often random, motion of molecules. A quantitative idea of the transfer of heat energy is obtained by noting that 1 kg of water releases about 42000 J of energy when it cools by10 °C.
7.2Chemical energy:
One of the greatest technical achievements of humankind occurred when we discovered how to ignite and control fire. We learnt to rub two flint stones together (mechanical energy), got them to heat up and to ignite a heap of dry leaves (chemical energy), which then provided sustained warmth. A matchstick ignites into a bright flame when struck against a specially prepared chemical surface. The lighted matchstick, when applied to a firecracker, results in a spectacular display of sound and light. Chemical energy arises from the fact that the molecules participating in the chemical reaction have different binding energies. A stable chemical compound has less energy than the separated parts. A chemical reaction is basically a rearrangement of atoms. If the total energy of the reactants is more than the products of the reaction, heat is released and the reaction is said to be an exothermic reaction. If the reverse is true, heat is absorbed and the reaction is endothermic. Coal consists of carbon and a kilogram of it when burnt releases about 3 × 107 J of energy. Chemical energy is associated with the forces that give rise to the stability of substances. These forces bind atoms into molecules, molecules into polymeric chains, etc. The chemical energy arising from the combustion of coal, cooking gas, wood and petroleum is indispensable to our daily existence.
7.3 Electrical energy:
The flow of electrical current causes bulbs to glow, fans to rotate and bells to ring. There are laws governing the attraction and repulsion of charges and currents, which we shall learn later. Energy is associated with an electric current. An urban Indian household consumes about 200 J of energy per second on an average.
LAWS OF MOTION
MOTION IN A PLANE
MOTION IN A STRAIGHT LINE
HIMANSHU CHAUDHARY
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