# Work, Energy and Power

### Work done

Work done by a force is the product of the forces and the distance moved in the direction of the force.
SI unit: joule (J) – the SI unit of energy.
1 J = 1 N m

\begin{aligned} W=F\times s \end{aligned}

Work done must be calculated by using the component of the force that is parallel to the direction of the displacement.
If the two directions are the same, work is done on the object and the objects gains energy.
If the two directions are opposite, then energy is lost through work done(typically as friction or air resistance)

### Work done by a gas

Work done by a gas equals to the pressure it exerts over the cross-sectional area and the displacement the piston moves.

\begin{aligned} F&=p\times A\\W&=p\times A\times s\\&=p\times \Delta V \end{aligned}

### Gravitational potential energy

Work is done by you during lifting. Object gains height when work is done against gravity. Gravitational potential energy increases as an object moves to a higher ground.

### Other forms of potential energy

There are many forms of potential energy. Some examples are below:

1. Electrical potential energy – capacitor. A capacitor stores charge which provides the capacitor the ability to produce current. A battery does not store electrical potential energy. Instead, it is stored as chemical energy.
2. Elastic potential energy – spring
3. Chemical potential energy – food, battery. Usually, chemical energy allows the substance to burn for a period of time, such as most organic compounds.
4. Nuclear energy – radioactive nuclei

### Kinetic energy

Kinetic energy is the energy due to motion. A moving object contains kinetic energy. To derive the equation of kinetic energy, we first assume an object is at rest. A constant force does work on the object, causing it to accelerate.

\begin{aligned} v^2&=u^2+2as\\&=2as \text{ since the object is initially at rest} \end{aligned}

Multiplying $\frac{1}{2}m$ on both sides,

\begin{aligned} \frac{1}{2}mv^2&=mas\\&=Fs \end{aligned}

Hence, work done is converted into kinetic energy of $\frac{1}{2}mv^2$ .

### Gravitational PE and Kinetic energy Transformation

When a high object falls to a lower height, the GPE is converted to KE. The increase in KE cause the object to increase in speed. The decrease in height should always be calculated based on the drop in vertical height.

### Efficiency

Efficiency is the ratio of the useful output energy to the total input energy expressed as a percentage.

\begin{aligned} \text{efficiency}=\frac{\text{useful output energy}}{\text{total input energy}} \times 100\% \end{aligned}

It is important for you to determine what makes the useful output energy in an energy transformation situation. Below are some situations of energy transformations. Suggest the useful output energy and the energy/energies that are lost.

1. A luggage being delivered from the ground floor to the first floor.
2. A man turning a generator to produce electricity.
3. Water flowing through a hydroelectric damp.
4. Wind flowing through a windmill.

# Energy of a Simple Harmonic Oscillator

### Energy changes in according to displacement

The kinetic energy of a simple harmonic oscillator is

\begin{aligned} E_\text{kinetic} &= \frac{1}{2}m \omega^2(x_o^2 - x^2) \end{aligned}

In a SHM, the oscillator’s kinetic energy and potential energy always changes from maximum to zero throughout the oscillations. However, at all time, the total energy of the oscillator is constant. This value can be obtained by calculating the maximum kinetic energy of the system:

\begin{aligned} E_\text{total} &= \frac{1}{2}mv_o ^2\\&=\frac{1}{2}m \omega^2 x_o^2 \end{aligned}

To find out the potential energy of a simple harmonic oscillator,

\begin{aligned} E_\text{potential} &= E_\text{total} - E_\text{kinetic}\\&=\frac{1}{2}m \omega^2x_o^2 - \frac{1}{2}m \omega^2(x_o^2 - x^2)\\&=\frac{1}{2}m\omega^2 x^2\end{aligned}

The energy changes of an SHM oscillator changes in a sinusoidal pattern.

### Energy changes according to time

From the velocity equation, the kinetic energy is

\begin{aligned} E_\text{kinetic}&=\frac{1}{2}mv^2\\&=\frac{1}{2}mv_o ^2 \cos^2{\omega t}\end{aligned}

The potential energy is the difference between the total energy and the kinetic energy,

\begin{aligned} E_\text{potential}&=\frac{1}{2}mv_o ^2 - \frac{1}{2}mv_o ^2 \cos^2{\omega t}\\&=\frac{1}{2}mv_o^2(1-\cos^2{\omega t})\\&=\frac{1}{2}mv_o \sin^2{\omega t} \end{aligned}