"th Class: 11Unit:- work, power, energy Subject - PHYSICS? Work:-Work done W is defined as the dot product of force F and displacement s.Here ? is the angle betweenand .° ° Work done by the force is positive if the angle between force and displacement is acute (0 <?<90 ) ascos ? is positive. This signifies, when the force and displacement are in same direction, work done ispositive. This work is said to be done upon the body.° ? When the force acts in a direction at right angle to the direction of displacement (cos90 = 0), no work isdone (zero work). ° ° ? Work done by the force is negative if the angle between force and displacement is obtuse (90 <?<180 )ascos ? is negative. This signifies, when the force and displacement are in opposite direction, work done isnegative. This work is said to be done by the body.? Work done by a variable force:-If applied force F is not a constant force, then work done by this force in moving the body fromposition A to B will be, Here ds is the small displacement.? Units:The unit of work done in S.I is joule (J) and in C.G.S system is erg.1J = 1 N.m , 1 erg = 1 dyn.cm7 ? Relation between Joule and erg:- 1 J = 10 erg? Power:-The rate at which work is done is called power and is defined as,P = W/t = F.s/v = F.vHere s is the distance and v is the speed.? Instantaneous power in terms of mechanical energy:- P = dE/dt? Units: The unit of power in S.I system is J/s (watt) and in C.G.S system is erg/s.? Energy:-1) Energy is the ability of the body to do some work. The unit of energy is same as that of work.2) Kinetic Energy (K):- It is defined as,2 K= ½ mvHere m is the mass of the body and v is the speed of the body.? Potential Energy (U):- Potential energy of a body is defined as, U = mgh Here, m is the mass of the body, g is the free fall acceleration (acceleration due to gravity) and h is theheight.? Gravitational Potential Energy:- An object’s gravitational potential energy U is its mass mtimes theacceleration due to gravity g times its height h above a zero level.In symbol’s,U = mgh? Relation between Kinetic Energy (K) and momentum(p):-2 K = p /2m? If two bodies of different masses have same momentum, body with a greater mass shall have lesserkinetic energy.? If two bodies of different mass have same kinetic energy, body with a greater mass shall have greatermomentum.? For two bodies having same mass, the body having greater momentum shall have greater kinetic energy.? Work energy Theorem:- It states that work done on the body or by the body is equal to the net change inits kinetic energy .? For constant force,2 2 W = ½ mv – ½ mu= Final K.E – Initial K.E? For variable force,? ? Law of conservation of energy:- It states that, “Energy can neither be created nor destroyed. It can beconverted from one form to another. The sum of total energy, in this universe, is always same”.? The sum of the kinetic and potential energies of an object is called mechanical energy. So, E = K+U? In accordance to law of conservation of energy, the total mechanical energy of the system always remainsconstant.2 So, mgh + ½ mv = constantIn an isolated system, the total energy E of the system is constant.total So, E = U+K = constantOr, U+K = U +Ki i f f Or, ?U = -?KSpeed of particle v in a central force field:v = v2/m [E-U(x)]? Conservation of linear momentum:-? In an isolated system (noexternal force ( F = 0)), the total momentum of the system before collision would be equal to totalext momentum of the system after collision.So, p = pf i ? Coefficient of restitution (e):- It is defined as the ratio between magnitude of impulse during period ofrestitution to that during period of deformation.e = relative velocity after collision / relative velocity before collision= v – v /u – u2 1 1 2Case (i) For perfectly elastic collision, e = 1. Thus, v – v = u – u . This signifies the relative velocities of2 1 1 2 two bodies before and after collision are same.Case (ii) For inelastic collision, e<1. Thus, v – v < u – u . This signifies, the value of e shall depend2 1 1 2 upon the extent of loss of kinetic energy during collision.Case (iii) For perfectly inelastic collision, e = 0. Thus, v – v =0, or v = v . This signifies the two bodies2 1 2 1 shall move together with same velocity. Therefore, there shall be no separation between them.? Elastic collision:- In an elastic collision, both the momentum and kinetic energy conserved.? One dimensional elastic collision:-? After collision, the velocity of two body will be,v = (m -m / m +m )u + (2m / m +m )u1 1 2 1 2 1 2 1 2 2 andv = (m -m / m +m )u + (2m / m +m )u2 2 1 1 2 2 1 1 2 1 Case:IWhen both the colliding bodies are of the same mass, i.e., m = m , then,1 2 v = u and v = u1 2 2 1 Case:IIWhen the body B of mass m is initially at rest, i.e., u = 0, then,2 2 v = (m -m / m +m )u and v = (2m / m +m )u1 1 2 1 2 1 2 1 1 2 1 (a) Whenm <<m , then, v = u and v = 2u2 1 1 1 2 1(b) Whenm =m , then, v =0and v = u2 1 1 2 1 (c) Whenm >>m , then, v = -u and v will be very small.2 1 1 1 2 ? Inelastic collision:- In an inelastic collision, only the quantity momentum is conserved but not kineticenergy.v = (m u +m u )/(m +m )1 1 2 2 1 2 and2 2 2 loss in kinetic energy, E = ½ m u + ½ m u - ½ (m + m )v1 1 2 2 1 2 or,2 2 2 E= ½ (m u + m u ) – ½ [(m u + m u )/( m + m )]1 1 2 2 1 1 2 2 1 2 2 = m m (u -u ) / 2( m + m )1 2 1 2 1 2 ? Points to be Notice:-(i) The maximum transfer energy occurs if m = m1 2 (ii) If K is the initial kinetic energy and K is the final kinetic energy of mass m , the fractional decreasei f 1 in kinetic energy is given by,2 2 K – K / K = 1- v /ui f i 1 1 Further, if m = nm and u = 0, then,2 1 2 2 K – K / K = 4n/(1+n)i f i ? Conservation Equation:(i)Momentum – m u +m u = m v +m v1 1 2 2 1 1 2 2 2 2 2 2 (ii) Energy – ½ m u + ½ m u = ½ m v + ½ m v1 1 2 2 1 1 2 2 ? Conservative force (F):- Conservative force is equal to the negative gradient of potential Vof the field ofthat force. This force is also called central force.So,F = - (dV/dr)? The line integral of a conservative force around a closed path is always zero. So,? Spring potential energy (E ):- It is defined as,s 2 E = ½ kxs Here k is the spring constant and x is the elongation.? Equilibrium Conditions:(a) Condition for equilibrium, dU/dx = 0(b) For stable equilibrium,U(x) = minimum,dU/dx = 0,2 2 d U/dx = +ve(c) For unstable equilibrium,U(x) = maximum dU/dx = 02 2 d U/dx = -ve(d) For neutral equilibrium,U(x) = constantdU/dx = 0 2 2 d U/dx = 0? UNITS AND DIMENSIONS OF WORK, POWER AND ENERGYWork and Energy are measured in the same units. Power, being the rate at which work is done, ismeasured in a different unit.Quantity andUnits/Dimensions Work (Energy) Power2 -2 2 -3 Dimension ML T ML TAbsolute MKS Joule WattFPS ft-Poundal ft-poundal/secCGS erg Erg/sec.Gravitational MKS kg-m Kg-m/secFPS ft-lb ft-lb/sec.CGS gm-cm Gm-cm/secPractical kwh, eV, cal HP, kW, MW(Other)? Conversions between Different Systems of Units5 2 7 1 Joule = 1 Newton ´ 1 m = 10 dyne ´ 10 cm = 10 erg 7 1 watt = 1 Joule/ sec = 10 erg/sec.3 3 1 kwh= 10 watt ´ 1 hr= 10 watt ´ 3600 sec 6 = 3.6 ´ 10 Joule1HP = 746 watt.6 1 MW = 10 watt.1 cal = 1 calorie = 4.2 Joule-19 1eV = "e" Joule= 1.6 ´ 10 Joule(e = magnitude of charge on the electron in coulombs) "