# Work, Energy and Power

Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force

W = F s cos θ

Negative work is said to be done by F if x or its compo. is anti-parallel to F
If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by “counting the squares”. }

By Principle of Conservation of Energy,

Work Done on a system = KE gain + GPE gain + Work done against friction}

Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s.

Since F is constant, acceleration is constant,

Therefore, using the equation:

v2 = u2 +2as,
as = 12 (v2 - u2)

Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v,

 The kinetic energy, EK = Work done by the force F = Fs = mas = ½ m (v2 - u2)

Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field.

Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them.

Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them.

The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by:
F = - dU / dx.

Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg.

 The change in potential energy of the mass = Work done by the force F = F s = F h = m g h

Efficiency: The ratio of (useful) output energy of a machine to the input energy.

 ie = Useful Output Energy x100% = Useful Output Power x100% Input Energy Input Power

Power {instantaneous} is defined as the work done per unit time.

 P = Total Work Done = W Total Time t

Since work done W = F x s,

 P = F x s = Fv t
• for object moving at const speed: F = Total resistive force {equilibrium condition}
• for object beginning to accelerate: F = Total resistive force + ma