# 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:

v^{2} = u^{2} +2as,

as = 12 (v^{2} - u^{2})

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

The kinetic energy, E_{K} |
= Work done by the force F = Fs = mas = ½ m (v ^{2} - u^{2}) |

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 h_{1} to a height h_{2}. Since the object moves up at a constant speed, F is equal to mg.

The in potential energy of the masschange |
= 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**