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WORK, ENERGY AND POWER 1 Chapter 05 Work, Energy And Power 1. Work Introduction to Work: In Physics, work stands for ‘mechanical work’. Work is said to be done by a force when the body is displaced actually through some distance in the direction of the applied force. However, when there is no displacement in the direction of the applied force, no work is said to be done, i.e., work done is zero, when displacement of the body in the direction of the force is zero. Suppose a constant force F  acting on a body produces a displacement s  in the body along the positive x-direction, as shown in the figure Fig.5.1 If  is the angle which force makes with the positive x– direction of the displacement, then the component of in the direction of displacement is (F cos  ). As work done by the force is the product of component of force in the direction of the displacement and the magnitude of the displacement, W Fcos s    ... 1  If displacement is in the direction of force applied,   0 . Then from (1), W = (F cos 0°) s = F s Equation (1) can be rewritten as W F.s  ... 2    Thus, work done by a force is the dot product of force and displacement. In terms of rectangular component, F  and s  may written as x y z ˆ ˆ ˆ F F i F j F k     and ˆ ˆ ˆ s xi yj zk     From (2), W F.s     x y z    W F i F j F k . xi yj zk ˆ ˆ ˆ ˆ ˆ ˆ      W x F y F zF    x y z Obviously, work is a scalar quantity, i.e., it has magnitude only and no direction. However, work done by a force can be positive or negative or zero. Note: Work done is positive, negative or zero depending upon the angle between force & displacement 1.1. Dimensions and Units of Work As work = force × distance   W M L T L 1 1 2   W = (M1 L1 T–2 ) × L W M L T 1 2 2      This is the dimensional formula of work. The units of work are of two types: 1. Absolute units 2. Gravitational units (a) Absolute unit 1. Joule. It is the absolute unit of work in SI. Work done is said to be one joule, when a force of one newton actually moves a body through a distance of one metre in the direction of applied force. From W Fscos   1 joule = 1 newton × 1 metre × cos 0° = 1 N–m 2. Erg. It is the absolute unit of work in cgs system. Work done is said to be one erg, when a force of one dyne actually moves a body through a distance of one cm in the direction of applied force. From W Fscos   5 2 1erg 1dyne 1cm cos0 10 N 10 m 1          7 1erg 10 J   (b) Gravitational units These are also called the practical units of work. 1. Kilogram-metre (kg–m). It is the gravitational unit of work in SI. Work, Energy and Power
10 WORK, ENERGY AND POWER Work done is said to be one kg–m, when a force of 1 kgf moves a body through a distance of 1 m in the direction of the applied force. From W Fcos   1 kg–m = 1kgf × 1 m × cos 0° = 9.8 N × 1 m = 9.8 joule, i.e., 1kg m 9.8J   2. Gram-centimetre (g-cm). It is the gravitational unit of work in cgs system. Work done is said to be one g-cm, when a force of 1g f moves a body through a distance of 1 cm in the direction of the applied force. From W Fscos   1 g-cm = 1 g f × 1 cm × cos 0° 1 g-cm = 980 dyne × 1 cm × 1 1g cm 980erg   1.2. Nature of Work Done Although work done is a scalar quantity, its value may be positive, negative or even zero, as described below: (a) Positive work As W = F.s Fscos      when q is acute (< 90°), cos q is positive. Hence, work done is positive. For example: When a body falls freely under the action of gravity,   0 , cos cos 0 1      . Therefore, work done by gravity on a body falling freely is positive. (b) Negative work As W = F. s Fscos     \ When q is obtuse (> 90°), cos q is negative. Hence, work done is negative. For example: When a body is thrown up, its motion is opposed by gravity. The angle  between gravitational force and the displacement is 180°. As cos cos180 1      therefore, work done by gravity on a body moving upwards is Note negative. Fig.5.2 (c) Zero work When force applied F  or the displacement s  or both are zero, work done W = F s cos q is zero. Again, when angle q between F  and s  is 90°, cos cos90 0     . Therefore, work done is zero. For example: When we push hard against a wall, the force we exert on the wall does no work, because s  = 0. However, in this process, our muscles are contracting and relaxing alternately and internal energy is being used up. That is why we do get tired. 1.3. Work done by a Variable Force If the force is variable then the work done is   xB xA W F x .dx   Fig.5.3 W Area ABCDA  Hence, work done by a variable force is numerically equal to the area under the force curve and the displacement axis. Note: NOTE: Energy of a body is defined as the capacity or ability of the body to do the work Work done is equal to energy consumed. D C F (x) x O A B Work (b) Work, Energy and Power
WORK, ENERGY AND POWER 11 2. Kinetic Energy Introduction to Kinetic Energy: The kinetic energy of a body is the energy possessed by the body by virtue of its motion. For example: (i) A bullet fired from a gun can pierce through a target on account of kinetic energy of the bullet. (ii) Windmills work on the kinetic energy of air. (iii) For example, sailing ships use the kinetic energy of wind. (iv) Water mills work on the kinetic energy of water. For example, fast flowing stream has been used to grind corn. (iv) A nail is driven into a wooden block on account of kinetic energy of the hammer striking the nail. Formula for Kinetic Energy 1 2 K.E. of body m v 2  2.1. Relation Between Kinetic Energy and Linear Momentum Let m = mass of a particle, v   velocity of the particle. Linear momentum of the particle, p mv    and K.E. of the particle   1 1 2 2 2 mv m v 2 2m   2 p K.E 2m   This is an important relation. It shows that a particle cannot have K.E. without having linear momentum. The reverse is also true. Further, if p = constant, 1 K.E m  This is shown in figure (a) If K.E. = constant, 2 p m or This is shown in figure (b). If m = constant, 2 p K.E  or This is shown in figure (c) Fig.5.4 3. Work Energy Theorem According to this principle, work done by net force in displacing a body is equal to change in kinetic energy of the body. Thus, when a force does some work on a body, the kinetic energy of the body increases by the same amount. Conversely, when an opposing (retarding) force is applied on a body, its kinetic energy decreases. The decrease in kinetic energy of the body is equal to the work done by the body against the retarding force. Thus, according to work energy principle, work and kinetic energy are equivalent quantities. Proof: To prove the work-energy theorem, we confine ourselves to motion in one dimension. Suppose m = mass of a body, u = initial velocity of the body, F = force applied on the body along it’s direction of motion, a = acceleration produced in the body, v = final velocity of the body after t second. Work, Energy and Power