# 8.1 Energy

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## External and internal forces

In Grade 10, you saw that mechanical energy was conserved in the absence of external forces. It is important to know whether a force is an internal force or an external force in the system, because this is related to whether the force can change an object's total mechanical energy when it does work on an object.

When an external force (for example friction, air resistance, applied force) does work on an object, the total mechanical energy (KE + PE) of that object changes. If positive work is done, then the object will gain energy. If negative work is done, then the object will lose energy. The gain or loss in energy can be in the form of potential energy, kinetic energy, or both. However, the work which is done is equal to the change in mechanical energy of the object.

## Investigations : external forces

We can investigate the effect of external forces on an object's total mechanical energy by rolling a ball along the floor from point A to point B.

Find a nice smooth surface (e.g. a highly polished floor), mark off two positions, A and B, and roll the ball between them.

The total mechanical energy of the ball, at each point, is the sum of its kinetic energy (KE) and gravitational potential energy (PE):

$\begin{array}{ccc}\hfill {E}_{\mathrm{total},\mathrm{A}}& =& {\mathrm{KE}}_{\mathrm{A}}+{\mathrm{PE}}_{\mathrm{A}}\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}+mg{h}_{A}\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}+mg\left(0\right)\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}\hfill \end{array}$
$\begin{array}{ccc}\hfill {E}_{\mathrm{total},\mathrm{B}}& =& {\mathrm{KE}}_{\mathrm{B}}+{\mathrm{PE}}_{\mathrm{B}}\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}+mg{h}_{B}\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}+mg\left(0\right)\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}\hfill \end{array}$

In the absence of friction and other external forces, the ball should slide along the floor and its speed should be the same at positions A and B. Since there are no external forces acting on the ball, its total mechanical energy at points A and B are equal.

$\begin{array}{ccc}\hfill {v}_{A}& =& {v}_{B}\hfill \\ \hfill \frac{1}{2}m{v}_{A}^{2}& =& \frac{1}{2}m{v}_{B}^{2}\hfill \\ \hfill {E}_{\mathrm{total},\mathrm{A}}& =& {E}_{\mathrm{total},\mathrm{B}}\hfill \end{array}$

Now, let's investigate what happens when there is friction (an external force ) acting on the ball.

Roll the ball along a rough surface or a carpeted floor. What happens to the speed of the ball at point A compared to point B?

If the surface you are rolling the ball along is very rough and provides a large external frictional force, then the ball should be moving much slower at point B than at point A.

Let's now compare the total mechanical energy of the ball at points A and B:

$\begin{array}{ccc}\hfill {E}_{\mathrm{total},\mathrm{A}}& =& {\mathrm{KE}}_{\mathrm{A}}+{\mathrm{PE}}_{\mathrm{A}}\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}+mg{h}_{A}\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}+mg\left(0\right)\hfill \\ & =& \frac{1}{2}m{v}_{A}^{2}\hfill \end{array}$
$\begin{array}{ccc}\hfill {E}_{\mathrm{total},\mathrm{B}}& =& {\mathrm{KE}}_{\mathrm{B}}+{\mathrm{PE}}_{\mathrm{B}}\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}+mg{h}_{B}\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}+mg\left(0\right)\hfill \\ & =& \frac{1}{2}m{v}_{B}^{2}\hfill \end{array}$

However, in this case, ${v}_{A}\ne {v}_{B}$ and therefore ${E}_{\mathrm{total},\mathrm{A}}\ne {E}_{\mathrm{total},\mathrm{B}}$ . Since

$\begin{array}{ccc}\hfill {v}_{A}& >& {v}_{B}\hfill \\ \hfill {E}_{\mathrm{total},\mathrm{A}}& >& {E}_{\mathrm{total},\mathrm{B}}\hfill \end{array}$

Therefore, the ball has lost mechanical energy as it moves across the carpet. However, although the ball has lost mechanical energy, energy in the larger system has still been conserved. In this case, the missingenergy is the work done by the carpet through applying a frictional force on the ball. In this case the carpet is doing negative work on the ball.

When an internal force does work on an object by an (for example, gravitational and spring forces), the total mechanical energy (KE + PE) of that object remains constant but the object's energy can change form. For example, as an object falls in a gravitational field from a high elevation to a lower elevation, some of the object's potential energy is changed into kinetic energy. However, the sum of the kinetic and potential energies remain constant. When the only forces doing work are internal forces, energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical energy is conserved.

Which equation do u wanna balance
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Thank you Christina. This is very helpful 😀👍👍💯
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💯💯
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