 # Potential and Kinetic Energy

## Potential Energy

Around all matter (mass) there is a gravitational field. The gravitational force between two masses is the product of both the masses. This force gives mass a 'potential' to move. When a rocket is fired into space, chemical energy is converted to kinetic, which is in turn converted to gravitational potential as it goes higher.

What happens when you climb up a lot of stairs? You get hot and tired, because you are burning a lot of energy.

Where does this energy come from, and what happens to it?

Your cellular respiration is 'burning' the carbohydrates you ingested as food. Some of this is used for bodily functions, such as heart operation and digestion, and some is lost as heat. But a large part of your respiration is used to generate the energy you need for motion. That is why we exercise to lose weight.

Why does it take more energy to climb stairs than walk along a flat path? Well, the answer is that when you walk along a flat path, you are not 'gaining' any other form of energy. But when you walk up stairs you are gaining gravitational potential energy - the energy of height in a gravitational field. A ski-lift converts electrical energy into gravitational potential energy when it lifts skiers to the top of a ski-slope. The skiers then convert this potential energy to movement energy, known as 'kinetic energy', when they race down the hill. Elastic potential in a watchspring provides the energy to drive the clockwork mechanism

Potential energy is also the energy stored in a spring. An example is an old mechanical clock, which stores the energy from the winding of the spring, released over a day or so to drive the mechanism of the clock or watch.

All machines involve the conversion of energy from one form to another. No conversion is ever 100% efficient. Some energy is also lost to heat and sound.

### The Maths

The gain in potential energy, Ep, of a mass, m, as it rises h metres in the Earth's gravitational field, with acceleration g, can be calculated from the equation:

\$\$E_p = mgh\$\$

and has the unit of Nm, or newton-metre, which is equivalent to a joule, J, the S.I. unit for energy.

The energy put in to the change in potential energy is the work done: \$W = ΔE_p\$

# Kinetic Energy

Gravitational potential is used by skiers to speed down a hill. Why do we gain speed when we come downhill, but have to work hard to get back up the hill? The Law of Conservation of Energy states that energy cannot be created or destroyed, only converted. We eat food, and use its energy to ride a bicycle up a hill, which increases our gravitational energy. When we come back down, we lose this gravitational energy and gain kinetic energy.

It is interesting to think that when we use a skilift, a part of its electrical energy is used to increase our gravitational potential. This is then converted to kinetic energy when we later ski past the lift station. We do not usually think of it, but skiers are enjoying 'electrical' energy.

### Efficiency

The same energy laws tell us that no energy conversion is perfect. This means that not all the energy can be converted to kinetic. This does not mean that the other energy is 'lost'. It is converted to some other form - in the case of the skier there is still some friction, against the snow and the air, which converts a part of the energy to heat and sound.

The degree to which a machine or system can convert energy to the desired, or 'useful' energy form, is known as the efficiency of the machine.

This 'imperfect' conversion is the reason why 'perpetual motion' machines are impossible. A machine converts energy from one form to another, but no machine can be set in motion and continue forever - there will always be some loss and dispersion of the energy to the surrounding environment. The gravitational potential energy in a large body of water can be used to generate electricity in a hydropower dam through its kinetic energy as it falls down the slope.

Water in a hydro-electric dam has gravitational potential energy. When it is allowed to fall through pipes this potential energy is converted to kinetic energy, which is converted to electrical energy by turbines. This skateboarder's kinetic energy reduces to zero at the top of his flight, where his potential energy is at maximum.

A good example of the conversion of gravitational potential and kinetic energy is a skateboarder on a half-pipe. He loses a little energy due to friction, but returns to almost the same height as he started. He can maintain his original height or even increase it by adding energy by pushing down on the board, causing a reaction force from the ground on the board. But this is added energy to the system, not energy from the original gravitational energy at the top of the half-pipe.

### The Maths

Kinetic energy can be calculated from the equation:

\$\$E_k = ½mv^2\$\$

where Ek is the kinetic energy, m is the mass, and v is the velocity.

and has the unit of joule, the S.I. unit for energy.

From this equation, it can be seen that the energy increases linearly with the mass, but to the square of the velocity. Double the speed and you quadruple the kinetic energy.

### Example A bullet has a mass of 80g. It is fired at 800 m/s. What is its kinetic energy? What was the chemical energy released by the explosion?

Answer: Ek = ½mv2 = 0.5 x 0.08 kg x 8002 = 25600 Joules = 25.6 kJ

The explosion of the charge in the bullet provided the kinetic energy above. However, the laws of energy conservation say that no conversion of energy from one form to another is 100%. There was also a loud noise and heat generated. So the chemical energy released was considerably more than 25.6 kJ - probably around 50 kJ, depending on the type of bullet and weapon design.

# Roller Coaster

This case study demonstrates how the principles of conservation of energy and transformations of energy with efficiency are used in engineering to make calculations to understand how a structure will behave, and how safe it is. A roller coaster wagon experiences constantly changing potential and kinetic energies throughout its journey

## Conservation of Energy

A fundamental principle of physics and engineering is that the total energy of a closed system remains constant. Of course, a roller coaster will lose energy continuously, due to friction on the rails, producing noise and heat, as well as air resistance. However, the loss can be calibrated: for example, for every x m, there is ten per cent loss of energy.

Engineers design to specifications. This means they are given a list of requirements that the machine or structure must meet. These will include factors such as load bearing (the total amount of stress the structure or machine can take before suffering damage or collapse). In the case of the roller coaster, efficiency of energy loss and maximum g-force experienced by the passengers are the specifications.

## Roller coaster dynamics

What happens when a roller coaster leaves the peak of its track? It starts with zero velocity and heads downhill. The total energy of the 'system' therefore is the potential energy it has at the top: no extra energy is added at any stage. At the lowest point of the track, all of the potential energy has been transformed to kinetic energy, and the wagon is moving at it fastest.

The general equation for the energy at any moment is:

\$\$E_{total} = E_p + E_k\$\$

At the top of the highest peak, the kinetic energy is zero so \$E_{total} = E_p + E_k = E_p_i = mgH\$, where H is the initial height.

At the lowest point in the track, the potential energy is zero so \$E_{total} = E_p + E_k = E_k\$.

At any other point on the track, the energy will be divided between Ep and Ek. The easiest way to calculate how much of each energy form there is, is to use the height h above the lowest point, and the formula: \$E_p = mgh\$

The kinetic energy is therefore: \$E_k = E_{total} - E_p = ½mv^2\$.

Notice that if the equation is written out in full: \$E_p_i = mgH = E_p + E_k = mgh + ½mv^2\$.

Or, \$mgH = mgh + ½mv^2\$. Cancelling the mass from both sides:

\$\$gH = gh + ½v^2\$\$

This can be rewritten as: \$½v^2 = g(H-h)\$

Therefore, the velocity at any point on the track at height h above the lowest point is:

\$\$v = √{2g(H-h)} = 4.43√{H-h}\$\$

### Loops In a loop, the wagon experiences centripetal force. The acceleration towards the centre of the loop is:

\$\$a_c = {v^2}/r\$\$

The speed of a wagon on entering a loop must not be too great, otherwise the g-forces will be dangerous, and not too low, otherwise the wagon and passengers may fall at the top.

The first loops in roller coasters, in France during the 1850s, in fact gave the passengers too much g-force, resulting in injuries. Modern roller coasters are not allowed to exceed about 5g.

To be safe, the diameter of the loop must be greater than 50% of the maximum descent prior to the loop.

### G-forces

G-forces is the measure of force in multiples of normal weight. We experience a g-force of 1 by standing in the Earth's gravitational field. When we stand in an elevator accelerating upwards at 1.0 m/s2 (or one-tenth of gravity), we experience a g-force of 1.1.

A jet pilot may experience 5 or 6 g-force when his plane exercises a violent manoeuvre. He may undergo as much as 10 g-force (10 g's) briefly when he fires rockets under his seat, to bail out in an emergency. This risks serious injury. Much more may be fatal to humans.

So, how many g's do we experience when we ride a loop on a roller coaster?

The first roller coasters to use loops, in France in the 1850s, had high g-forces, and did in fact cause injury to passengers. As a result, the loop was abandoned as a feature on roller coasters until the 1920s, when Coney Island engineers applied physics to design a safer coaster.

And now it is your turn: design a roller coaster to scale, using polystyrene half-tubes as the tracks. The wagon can be simulated roughly to scale by marbles. Include a loop and calculate the maximum g-force the marble experiences. To be safe, there should never be more than 4 g.

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