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22.4 Mechanical energy

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*}
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