EGR 280 Mechanics 12 – Work and Energy of Particles

Work and Energy Newton’s Second Law describes the vector relationship between force, acceleration and time. Work and Energy is a restatement of Newton’s Second Law, and describes the scalar relationship between force, position and speed. The work done by a force Let the change in the position of a particle be displacement = ∆r The work done by a force in moving a particle from point A to another point B is defined as U AB = F ∙ dr A B F

Special cases: 1. Work of a constant force in rectilinear motion: U AB = ∫ A B F ∙ dr = ∫ A B F cosθ dx = F cosθ (x B – x A ) = F cosθ ∆x 2. Work done by a particle’s weight U AB = ∫ A B F ∙ dr = ∫ A B -W dy = -W (y B – y A ) = -W ∆y F A B θ y A B yAyA yByB W

3.Work of the force exerted by a linear spring U AB = ∫ A B F ∙ dr = ∫ A B -kx dx = -½ k (x B 2 – x A 2 ) = ½ k (x A 2 – x B 2 ) L0L0 xAxA xBxB F = -kx

Kinetic Energy of a Particle Consider the motion of a particle in intrinsic coordinates. The component of any force applied to the particle in the normal direction does no work on the particle, since it is always perpendicular to the direction of motion. Therefore, all the work on the particle is done by the tangential force: F t = m dv/dt = m(dv/ds)(ds/dt) = mv(dv/ds) ∫ A B F t ds = ∫ A B mv dv U AB = ½ m(v B 2 – v A 2 ) Define the kinetic energy of the particle, T, as ½ mv 2, so that T A + U AB = T B U AB = T B - T A etet enen

Power Power is defined as the rate at which work is done: P = dU/dt = d/dt(U) = (F ∙ dr)/dt = F ∙ dr/dt P = F ∙ v 1 W = 1 J/s = 1 N-m/s 1 hp = 550 ft-lb/s = 746 W Efficiency is the ratio of the output power to the input power: η = (power output) / (power input)

Potential Energy We have seen that the work in moving a body against gravity is U AB = -W ∆y = -W (y B – y A ) = Wy A - Wy B The work done on the particle does not depend on the path taken between points A and B, it only depends on the heights of A and B above a datum at y=0. This function Wy is called the potential energy of the body with respect to gravity, or V g. Thus, U AB = V gA - V gB Likewise, if we define the potential energy of a linear spring as V e = ½ k x 2, then we can write the work done by the spring as U AB = V eA - V eB

Conservation of Mechanical Energy If the work done by all of the forces acting on a particle is independent of the path taken by the particle, then the force system is called conservative and the total mechanical energy, T+V, is conserved: U AB = V A – V B = T B – T A V A + T A = V B + T B If friction acts on the particle, then the total mechanical energy is not conserved and this expression can be written as V A + T A = V B + T B + energy lost