# Chapter IV Work and Energy

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Chapter IV Work and Energy
Work Done by a Constant Force The Work-Energy Theorem Work Done by a Varying Force or on Curved Path Power Gravitational Potential Energy and Elastic Potential Energy When Total Mechanical Energy is Conserved

A. Work Done by a Constant Force
Fx = F cos  x The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement: W = F  X W = F Cos  X F W x

B. The Work-Energy Theorem
W = F x = m a x v 2 = vo2 + 2 a x Kinetic energy  K = ½ m v2 W = K – Ko Wt =  K  work-kinetic energy theorem The net work done on a particle by a constant net force F acting on it equals the change in kinetic energy of the particle

C. Work Done by a Varying Force or on Curved Path
W = F  x W =  W = Limit  F  x  x  0 W =  F dx =  m a dx a = dv/dt = (dv/dx)(dx/dt) = (dv/dx) v W =  m a dx =  m (dv/dx) v dx W =  m dv v W = ½ m v 2 - ½ m vo2 = K – Ko Wt =  K W W x  x The net work done on a particle by the net force acting on it is equal to the change in the kinetic energy of the particle. By spring F = - k x By hand F = k x W =  F dx =  k x dx = ½ k x2 Elastic Potential Energy

D. Power P = Power (watt  W) W = work (joule  J)
The time rate of doing work is called power. P = Power (watt  W) W = work (joule  J) t = time (second  s) x = displacement (meter  m) v = velocity (m/s)

E. Gravitational Potential Energy and Elastic Potential Energy
W = F y = mg (h2 – h1) = m g h2 - m g h1 Gravitational potential energy = V = mgh W = V2 – V1 W =  V  work done by palm force W = -  V  work done by gravitation force WK = -  V h2 v h1 WK = work done by concervative force (J) g = acceleration of gravity (m/s2) m = mass(kg) V = Potential energy (J) WK = - (½ k x22 - ½ k x12 )  work done by spring force k = force constant of the spring(J/m2  N/m)

Conservative Force Conservative forces have two important properties: 1. A force is conservative if the work it does on a particle moving between any two points is independent of the path taken by the particle. 2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.)

E. When Total Mechanical Energy is Conserved
Wt = WNK + WK WNK = Wt – WK WNK =  K – (-  V) WNK = (K2 – K1) + (V2 – V1) WNK = work done by nonconcervative force (J) If WNK = 0, than K1 + V1 = K2 + V2 Mechanical energy  M = K + V M1 = M2  M = constant ½ m v 2 + mgh = constant  Conservation of Mechanical Energy

Increase (or decrease) in potential energy is accompanied by an equal decrease (or increase) in kinetic energy. The total mechanical energy of a system remains constant in any isolated system of objects that interact only through conservative forces.