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WORK, ENERGY AND POWER 73 5.2 NOTIONS OF WORK AND KINETIC to be proportional to the speed of the drop ENERGY:THEWORK-ENERGYTHEOREM but is otherwise undetermined. Consider Thefollowingrelationforrectilinearmotionunder a drop of mass 1.00 g falling from a height constant acceleration a has been encountered 1.00 km. It hits the ground with a speed of in Chapter 3, -1 2 2 50.0 m s . (a) What is the work done by the v − u = 2 as (5.2) gravitational force ? What is the work done where u and v are the initial and final speeds by the unknown resistive force? and s the distance traversed. Multiplying both Answer (a) The change in kinetic energy of the sides by m/2, we have drop is 1 2 1 2 1 mv − mu = mas = Fs (5.2a) ∆K = m v − 0 2 2 2 where the last step follows from Newton’s Second = 1 ×10 -3×50×50 Law. We can generalise Eq. (5.2) to three 2 dimensions by employing vectors = 1.25 J v − u = 2 a.d where we have assumed that the drop is initially at rest. Here a and d are acceleration and displacement vectors of the object respectively. Assum2ng that g is a constant with a value Onceagainmultiplyingbothsidesbym/2,weobtain 10 m/s , the work done by the gravitational force is, 1 1 mv − mu = m a.d = F.d (5.2b) W gmgh 2 2 = 10 ×10 ×10 3 The above equation provides a motivation for = 10.0 J the definitions of work and kinetic energy. The (b) From the work-energy theorem left side of the equation is the difference in the quantity ‘half the mass times the square of the ∆K = W +gW r speed’ from its initial value to its final value. We where W is the work done by the resistive force r call each of these quantities the ‘kinetic energy’, on the raindrop. Thus denoted by K. The right side is a product of the W r ∆K − W g displacement and the component of the force = 1.25 −10 along the displacement. This quantity is called = − 8.75 J ‘work’ and is denoted by W. Eq. (5.2b) is then is negative. ⊳ Kf− K i W (5.3) 5.3 WORK where K ind K arf respectively the initial and As seen earlier, work is related to force and the final kinetic energies of the object. Work refers displacement over which it acts. Consider a to the force and the displacement over which it constant force F acting on an object of mass m. acts. Work is done by a force on the body over The object undergoes a displacement d in the a certain displacement. positive x-direction as shown in Fig. 5.2. Equation (5.2) is also a special case of the work-energy (WE) theorem : The change in kinetic energy of a particle is equal to the work done on it by the net force. We shall generalise the above derivation to a varying force in a later section. ⊳Example 5.2 It is well known that a raindrop falls under the influence of the Fig. 5.2 An object undergoes a displacement d downward gravitational force and the under the influence of the force F. opposingresistiveforce. Thelatterisknown Reprint 2026-27