Work, energy and power

Energy is a fundamental concept and a basis upon which much of science is understood.

• Work done by a force is the product of the magnitude of the force $\rm (F)$ and the distance(s) travelled in the direction of the force:
  
  $\rm W = Fs \cos \theta$, where $\theta$ is the angle between the force and the direction of motion.
• Work is a scalar quantity.
  When the force is not constant the work can be calculated from the area under the force–distance graph.

As a spring is stretched the force increases from $0$ to $kx$.

$\mathrm{F_{average}} = ½ ~kx$

Average force $\times$ distance $= ½ ~kx \times = ½ ~k~x^2$

This is the area of the red shaded triangle.

This gives the expression for Elastic potential energy $= ½~k~ x^2$ (see below).

• Energy is the ability to do work. It is a scalar quantity.

• The Conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

• Kinetic energy is the energy due to movement:
  $\mathrm{E_K} = ½ ~mv^2$ (another formula for kinetic energy using momentum is $\mathrm{E_K} = p^2/2~m$
• Potential energy is energy due to position.
• Gravitational Potential energy increases when a body is moved a height h in a gravitational field:
  $\mathrm{E_P} = mgh$
• Elastic potential energy of a spring increases when it is stretched:
  $\mathrm{E_E} = ½ ~kx^2$ ($k$ is the spring constant).

• Power $\bf (P)$ is the rate at which a force does work (or rate at which energy is being consumed): $\rm i.e.$
  $\rm P = \Delta W/\Delta \mathcal t$
• Power is measured in joules per second, $\rm i.e.$ in watt $\rm (W)$. It is a scalar quantity.
• When a constant force $\rm F$ acting on a body moving with a speed $v$ along a straight line:
  $\rm P = \Delta W/\Delta \mathcal t = F \times \Delta \mathcal s/\Delta \mathcal t = F_{\mathcal v}$

• The energy output in transformations is not always all useful energy. Efficiency as the ratio of useful energy out to the total energy transferred.

Efficiency $=$ useful energy out / actual energy in

Efficiency $=$ useful power out / actual power in