Mechanics Tutorial 15 - Conservation of Energy

 Contents

Law of Conservation of Energy

The Law of Conservation of Energy states:

Energy is neither created nor destroyed; it is converted from one form to another.

At GCSE you would have done some energy chains, for example in a nuclear power station. • Nuclear energy is converted into heat.

• Heat boils water to steam

• Heat in the steam is converted into kinetic energy in the turbines

• Which is converted into electrical energy in the generator.  The above nonsense (including spelling mistake) is reproduced from a student's answer in a test!  Please be a good chap and don't write drivel like this.

Potential Energy

This term is often used in the context of gravitational potential energy.  If we lift an object of mass m against gravity, we are doing a job of work.

Work done = PE = weight × distance moved against gravity.

Work done:

W = Fs

Weight is a force:

F = mg

and the distance moved in the direction of the force is the vertical height difference:

s = Dh

The work done is the potential energy, therefore:

DEp = mgDh

 Question 1 Explain what each of the terms in the equation DEp = mg × Dh mean. Question 2 What is the potential energy of a 12 kg mass raised to a height of 25 m? Use g = 9.8 m s-2.  Give your answer to an appropriate number of significant figures. Gravitational potential energy is a linear relationship.  Therefore if we plot the gravitational potential energy against the height difference, the graph would be like this: In this case the graph shows direct proportionality. This only applies when we are considering the height difference, which we do almost all the time.  However, always be careful when you see something like. "...the crane raises a load from a height of 15 m to 25 m".  Then you have to work out the height difference - but it's not that hard!

If the energy was plotted against the absolute height, the graph would be linear, but NOT proportional. The potential energy relationship is only true when the gravity field is uniform.  This means that the object is close to the ground.  As we move away from the Earth, we find that the uniform model of the Earth's gravitational field breaks down, and we have to find a different solution.  This is discussed in Fields Tutorial 1.

Change in gravitational potential is the change in energy per unit mass (Fields Tutorial 2).   It is given the physics code DVg, and the units are Joule per kilogram (J kg-1). The change in gravitational potential (DVg) near the Earth's surface is linear, and is given by:

DVg = gDh

We often use the term potential energy in the context of gravitational potential energy.  There are other kinds of potential energy, for example, elastic potential energy in which energy is stored in objects under stress.  Objects that are under stress are those that are stretched (under tension), squashed (under compression) or twisted (under torque).  We will look at elastic potential energy, or elastic strain energy, in Materials Tutorial 2.

Other forms of potential energy include chemical potential energy from the bonds of reacting chemicals.  Energy is released as heat, that can be made to do work in an engine.

Kinetic energy is the ability to do work through motion.  If the motion is in a straight line, we call the kinetic energy translational

Kinetic energy is given by the equation:

kinetic energy (J) = 1/2 × mass (kg) × (speed (m s-1))2

In Physics Code: Suppose we use a force F to accelerate an object steadily from zero up to a speed v. The kinetic energy is the same as the work done.

Kinetic energy (J) = force (N) × distance moved (m) .......Eq 1

Since:

Force (N) = mass (kg) × acceleration (m s-2)

we can write:

Force (N) = mass (kg) × [change in speed (m s-1) ÷ time (s)]

In Physics code:

F = m × [(v - 0) ÷ t] = m × v/t

Now:

distance travelled = average speed × time

In Physics code:

d = [(v - 0) ÷ 2] × t = 1/2 v × t .........Eq 2

Combining equations 1 and 2:

Kinetic energy = F × d

Ek = m × v/t × 1/2 v × t

The t terms cancel out to give us:

Ek = 1/2 mv2

The ability to follow this derivation is the sort of thing that will distinguish the outstanding student (A*) from the excellent student (A).

 Question 3 Calculate the kinetic energy of a 4 kg shot-put thrown by an athlete at a speed of 15 m s-1. Note that we have talked about speed rather than velocity. The kinetic energy of an object is the same regardless of direction. We can say this, because of we have a negative velocity, the kinetic energy will still be positive, since minus times a minus is a plus.

Potential Energy converted to Kinetic Energy

If an object falls, the potential energy is turned into kinetic energy.  Then we combine the equations for Ep and Ek (conservation of energy):

Ep = Ek

mgDh = ½ mv2

mgDh = ½ mv2 [masses cancel]

Þ v2 = 2gDh

 Question 4 A coin is dropped from the viewing platform of an observation tower 80 m high.   How fast will it travel just before it hits the ground? Explain why you do not need the mass of the coin in the question above.
##### Many chemical energy resources such as fossil fuels have the advantage that their energy is in very concentrated form.  50 litres of diesel fuel can take a car 1000 km.  They have the two main disadvantages:

• They are non renewable.  Once the resources have been depleted, that’s it.

• They cause pollution.  Carbon dioxide is responsible for global warming.  Sulphur dioxide causes acid rain.

Electric cars do not have these problems.  However their range is limited by their batteries to about 200 – 300 km at best.  Electric cars are nothing new; they were around long before petrol cars.  However the limitations of the batteries have always been a problem.  Modern batteries give a greater capacity, but are very expensive.  It is proposed to replace all petrol and diesel vehicles by 2040.  However there is the small problem of infrastructure to allow every house to charge their electric vehicle - it costs money.  (If a government chickens out at the cost of erecting 3000 overhead power line masts to electrify the 200 km railway between Kettering and Sheffield, then what hope is there for them to wire fifteen million homes?)

Fossil fuels are primary energy sources.  A primary energy source is one that occurs naturally.  Examples include:

• Crude oil;

• Coal;

• Sunlight;

• Nuclear fuel.

Petrol for cars is refined from crude oil, and is described as a secondary energy source.  These renewable energy sources are secondary, with their primary driver being sunlight:

• Wind;

• Solar power;

• Hydro electric power;

• Wave power.

Tidal power gets its primary energy from the gravitational pulls from the Moon and the Sun.

Most renewable energy projects centre on the generation of electricity.   Electricity is a secondary source that is a particularly convenient to transmit, especially over long distances, and easy to use. It is flexible and can do jobs that other ways of transmission cannot do.

Each has its advantages and disadvantages, as shown in the table below:

 Renewable Source Advantage Disadvantage Hydro-electric Uses running water of which there is plenty. Renewable. Can be turned on and off quickly Usually needs a dam to block off a valley to make a reservoir. Wind Quiet Relatively inexpensive No emissions. Can be set up in any windy spot. Useless on a calm day Cannot work if the wind is too strong Often not welcomed by local people. Can disturb birds on migration. Solar Cells The intensity of the Sun is 500 W m-2 Huge amounts of energy can be harvested. Solar cells are expensive. Use scarce resources Inefficient. Need large panels Not that effective on dull days. Solar heating Suns rays can be concentrated with mirrors. Lots of heat in a small space. Needs large mirrors. Mirrors need to track the sun. Does not work well on a cloudy day. Potentially dangerous with intensely concentrated rays of light Solar panels Can be placed on any roof of a house. Make lots of hot water. Water needs to be stored. Ineffective on a cloudy day. Biomass New plants can be grown to replace the fuel Valuable agricultural land taken up for biomass crops. Low energy concentration of biomass fuels.
 Biogas Uses waste material such as rubbish and sewage.  Unlimited supply of these materials. People don’t like using it as they are put off by the thought of the source. Tidal Large amounts of power from tidal flows Will be there as long as the Moon’s there (and nobody is going to take it away). Environmental considerations, e.g. loss of habitat for wading birds. Power produced according to tides, not when people want it. Very expensive. Waves A lot of energy available in waves. Useful for countries with long coastlines. Machines need to be set up in hostile environments. Often damaged by storms. Geothermal Unlimited heat from the centre of the Earth On most places, deep drilling has to take place to get rocks of adequate temperature.  Steam can leak through cracks in the rocks. Best in volcanic areas.

A further discussion on wind turbines, hydroelectric power, tidal power, and pumped storage power stations can be found at Physics 6 Tutorial 12.

### Energy in Food

To do everything we need to do, we need energy.  We get this from our food.  Clearly if we don’t eat enough food, we don’t have much energy.  Prolonged lack of food leads to malnutrition with a variety of different health issues.

The converse of this is that people who eat too much get fat.  Excessive fatness leads to obesity, which can cause a very large number of health problems, including:

• Diabetes;

• Stroke;

• Heart attacks;

• Joint problems;

• Cancer.

Physicists are not exempt from this. Food energy is most commonly expressed in calories. Rather confusingly the food calorie is 1000 calories. In referring to calories, we will use kilocalories (kcal).

1 kilocalorie is the amount of energy required to raise the temperature of 1 kilogram of water by 1 Kelvin.

1 kcal = 4.2 kJ = 4200 J

Food energy is converted into two different kinds of energy:

• Heat energy to keep us warm and ready for action;

• Kinetic energy, produced when our muscles move.

If we do exercise we get hot as well us pumping iron.

An active young man requires about 2500 kcal of energy. The figure for a young woman is about 2000 kcal.

 Question 5 Convert these figures to kilojoules. A tutor has a mass of 80 kg and climbs the stairs from the first floor to the sixth floor of a college building, which has a total height of 21 m. Calculate the energy used in raising the tutor through that height. Convert your answer to kilocalories.  Use g = 9.81 N kg-1 The tutor likes flapjack, which has energy of 400 kcal for 100 g. How much flapjack will provide the tutor with the energy he needs as he goes up the stairs?

The answer to Question 7 will explain why, if you are trying to lose weight, doing a heavy session in the gym and then having a chocolate bar afterwards is counterproductive.

Not all energy is used by the body. Some food is not digestible, and forms the material produced when we sit on the lavatory.