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