Energy

Units is usually $1 J \ \text{(Joule)} = 1N\ \text{Newton} \cdot 1m\ \text{metre}$; or $1J = 1kg \cdot 1 m^{2} / 1 s^{2}$

Principal of conservation of energy

1. Energy cannot be created or destroyed
2. only transferred from one store to another
3. The total amount of energy in a system is constant

Work Done the amount of energy transferred when a force moves an object over a distance

$$\begin{array}{l} \text{Work done} = Fd \\ \text{\textbf{where}} \\ \quad \begin{array}{rl} F & \text{force in the direction of displacement in newtons} \\ d & \text{displacement in metres} \end{array} \end{array}$$

Power

The rate at which work is done or energy is transferred over time

$$\begin{array}{l} P = \frac{W}{t} \\ \text{\textbf{where}} \\ \quad \begin{array}{rl} P & \text{power in $W$ (watt)} \\ W & \text{work done in Joules} \\ t & \text{time taken in seconds} \end{array} \end{array}$$ $$\begin{array}{l} P = \frac{\Delta E}{t} \\ \text{\textbf{where}} \\ \quad \begin{array}{rl} \Delta E & \text{change in energy (or energy transferred) in Joules} \\ t & \text{time taken in seconds} \end{array} \end{array}$$

Instatenous power

\begin{align*} P &= \frac{W}{t} \\ &= \frac{Fd}{t} \\ &= F(\frac{d}{t}) \\ &= Fv \\ &\hspace{-5em} \rlap{\text{\textbf{where}}}\phantom{P} & \\ &\hspace{-5em} \rlap{\quad \begin{array}{rl} F & \text{force in Newtons} \\ v & \text{velocity or speed in metres per second} \end{array}}\phantom{P} & \end{align*}

Stores of energy

kinetic store

\begin{align*} E_k &= \frac{1}{2}mv^2 \\ &= \frac{1}{2}pQv^2 \\ & \hspace{-5em} \rlap{\text{\textbf{where}}}\phantom{E_k} & \\ & \hspace{-5em} \rlap{\quad \begin{array}{rl} m & \text{mass in Kilograms} \\ v & \text{velocity in metres per second} \\ p & \text{density of water in $Kg/m^3$} \\ Q & \text{volumetric flow rate in $m^3/s$} \\ \end{array}}\phantom{E_k} & \end{align*} $$\begin{align*} v &= \frac{Q}{A} \\ &\hspace{-5em} \begin{array}{l} \text{\textbf{where}} \\ \quad \begin{array}{rl} v & \text{fluid velocity in $ms^{-1}$} \\ Q & \text{volumetric flow rate in $m$} \\ A & \text{cross-sectional area in } \\ \end{array} \end{array} \end{align*}$$

gravitational potential store

$$\begin{align*} E_p &= mgh \\ &\hspace{-5em} \begin{array}{l} \text{\textbf{where}} \\ \quad \begin{array}{rl} m & \text{mass in Kilograms} \\ g & \text{acceleration due to gravity} \\ h & \text{heigh in metres} \end{array} \end{array} \end{align*}$$

elastic potential store

\begin{align*} E_e &= \frac{1}{2}kx^2 \\ & \hspace{-5em} \rlap{\text{\textbf{where}}}\phantom{E_e} & \\ & \hspace{-5em} \rlap{\quad \begin{array}{rl} k & \text{spring constant in newtons per metre ($N/m$)} \\ x & \text{extension in metres} \end{array}}\phantom{E_e} & \end{align*}

chemical potential store

Internal store

nuclear store

means of energy transfer

1. Mechanically -> force
2. Propagation of waves (eg sound waves) 
3. Heating (2 different temperatures) 
4. Electrical

$$\begin{align*} P &= J / s \\ &\hspace{-5em} \begin{array}{l} \text{\textbf{where}} \\ \quad \begin{array}{rl} P & \text{} R & \text{total energy or work done in joules ($J$)} \\ \end{array} \end{array} \end{align*}$$

Work Energy Theorem

the net work done by all forces acting on an object is equal to the change in its kinetic energy

$$\begin{align*} W_{\text{net}} &= \Delta E_k \\ \end{align*}$$

Efficiency

$$\begin{align*} \eta &= \frac{E_{\text{out}}}{E_{\text{in}}} \cdot 100\% \\ \end{align*}$$