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## 22.4 Mechanical energy (ESAHN)

Mechanical energy

Mechanical energy is the sum of the gravitational potential energy and the kinetic energy of a system.

Quantity: Mechanical energy ($${E}_{M}$$)         Unit name: Joule         Unit symbol: $$\text{J}$$

Mechanical energy, $${E}_{M}$$, is simply the sum of gravitational potential energy ($${E}_{P}$$) and the kinetic energy ($${E}_{K}$$). Mechanical energy is defined as:

${E}_{M}={E}_{P}+{E}_{K}$ ${E}_{M} = mgh + \frac{1}{2}m{v}^{2}$

You may see mechanical energy written as $$U$$. We will not use this notation in this book, but you should be aware that this notation is sometimes used.

## Worked example 6: Mechanical energy

Calculate the total mechanical energy for a ball of mass $$\text{0,15}$$ $$\text{kg}$$ which has a kinetic energy of $$\text{20}$$ $$\text{J}$$ and is $$\text{2}$$ $$\text{m}$$ above the ground.

### Analyse the question to determine what information is provided

• The ball has a mass $$m = \text{0,15}\text{ kg}$$

• The ball is at a height $$h = \text{2}\text{ m}$$

• The ball has a kinetic energy $${E}_{K} = \text{20}\text{ J}$$

### Analyse the question to determine what is being asked

We need to find the total mechanical energy of the ball

### Use the definition to calculate the total mechanical energy

\begin{align*} {E}_{M} & = {E}_{P}+{E}_{K} \\ & = mgh + \frac{1}{2}m{v}^{2} \\ & = mgh + 20 \\ & = \left(\text{0,15}\text{ kg}\right)\left(\text{9,8}\text{ m·s$^{-1}$}\right)\left(\text{2}\text{ m}\right) + \text{20}\text{ J} \\ & = \text{2,94}\text{ J} + \text{20}\text{ J} \\ & = \text{22,94}\text{ J} \end{align*}