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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

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**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

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**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

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**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

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**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)

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**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)

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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.)

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**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

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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.

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