Climbing bars

If you are going to climb a mountain, you will need a supply of energy. This is because your gravitational potential energy is greater at the top of the mountain than at the base. A good supply of energy would be some bars of chocolate. Each bar supplies $1200 kJ$. Suppose your weight is $600 N$ and you climb a $2000 m$ high mountain. The work done by your muscles is:

$work\,done\, = \,{F_s}\, = \,600 \times 2000 = 1200kJ$

So, one bar of chocolate should provide enough energy. Of course, in reality, it would not. Your body is inefficient. It cannot convert $100\% $ of the energy from food into gravitational potential energy. A lot of energy is wasted as your muscles warm up, you perspire and your body rises and falls as you walk along the path. Your body is perhaps only $5\% $ efficient as far as climbing is concerned, and you will need to eat 20 chocolate bars to get you to the top of the mountain. And you will need to eat more to get you back down again.

Many energy transfers are inefficient. That is, only part of the energy is transferred to where it is wanted.
The rest is wasted, and appears in some form that is not wanted (such as waste heat) or in the wrong place. You can determine the efficiency of any device or system using the following equation:

$efficiency\, = \,\frac{{useful\,output\,energy}}{{total\,input\,energy}} \times 100\% $

A car engine is more efficient than a human body, but not much more. Figure 5.15 shows how this can be represented by a Sankey diagram. The width of the arrow represents the fraction of the energy which is transformed to each new form. In the case of a car engine, we want it to provide kinetic energy to turn the wheels. In practice, $80\%$ of the energy is transformed into heat: the engine gets hot, and heat escapes into the surroundings. So the car engine is only $20\%$ efficient.

We have previously considered situations where an object is falling, and all of its gravitational potential energy changes to kinetic energy.
In Worked example 5, we will look at a similar situation, but in this case the energy change is not $100\%$ efficient.

WORKED EXAMPLE

5) Figure 5.16 shows a dam that stores water. The outlet of the dam is $20 m$ below the surface of the water in the reservoir. Water leaving the dam is moving at $16\,m\,{s^{ - 1}}$. Calculate the percentage of the gravitational potential energy that is lost when converted into kinetic energy.

Step 1: We will picture $1 kg$ of water, starting at the surface of the lake (where it has g.p.e., but no k.e.) and flowing downwards and out at the foot (where it has k.e., but less g.p.e.). Then:
$change\,in\,g.p.e.\,of\,water\,between\,surface\,and\,outflow\, = \,mgh\, = 1 \times 9.81 \times 20 = \,196J$
Step 2: Calculate the k.e. of $1 kg$ of water as it leaves the dam:
$\begin{array}{l}
k.e\,of\,water\,leaving\,dam\, = \frac{1}{2}m{v^2}\\
 = \frac{1}{2} \times 1 \times {(16)^2}\\
 = 128J
\end{array}$
Step 3: For each kilogram of water flowing out of the dam, the loss of energy is:
$loss = 196 − 128 = 68 J$
$\begin{array}{l}
percentage\,loss = \frac{{68}}{{196}} \times 100\% \\
 = 34.69\% \, \approx \,35\% 
\end{array}$
If you wanted to use this moving water to generate electricity, you would have already lost more than a third of the energy that it stores when it is behind the dam.

Conservation of energy

Where does the lost energy from the water in the reservoir go? Most of it ends up warming the water, or warming the pipes that the water flows through. The outflow of water is probably noisy, so some sound is produced.
Here, we are assuming that all of the energy ends up somewhere. None of it disappears. We assume the same thing when we draw a Sankey diagram. The total thickness of the arrow remains constant. We could not have an arrow which got thinner (energy disappearing) or thicker (energy appearing out of nowhere).
We are assuming that energy is conserved. This is a principle, known as the principle of conservation of energy, which we expect to apply in all situations.
Energy cannot be created or destroyed. It can only be converted from one form to another.
We should always be able to add up the total amount of energy at the beginning, and be able to account for it all at the end. We cannot be sure that this is always the case, but we expect it to hold true.
We have to think about energy changes within a closed system; that is, we have to draw an imaginary boundary around all of the interacting objects that are involved in an energy transfer.
Sometimes, applying the principle of conservation of energy can seem like a scientific fiddle. When physicists were investigating radioactive decay involving beta particles, they found that the particles after the decay had less energy in total than the particles before. They guessed that there was another, invisible particle that was carrying away the missing energy. This particle, named the neutrino, was proposed by the theoretical physicist Wolfgang Pauli in 1931. The neutrino was not detected by experimenters until 25 years later.
Although we cannot prove that energy is always conserved, this example shows that the principle of conservation of energy can be a powerful tool in helping us to understand what is going on in nature, and that it can help us to make fruitful predictions about future experiments.