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3.4.1.7 Work, energy and power

Energy transferred, $W=F\cos \theta$

rate of doing work = rate of energy transfer,$P=\frac{\Delta W}{\Delta t}=Fv$

Quantitative questions may be set on variable forces.

Significance of the area under a force–displacement graph.

$$\small efficiency=\frac{useful\: output\: power}{input\:power}$$

Efficiency can be expressed as a percentage.

3.4.1.8 Conservation of energy

Principle of conservation of energy.

$\Delta E_{p}=mg\Delta h$ and $E_{k}=\frac{1}{2}mv^{2}$

Quantitative and qualitative application of energy conservation to examples involving gravitational potential energy, kinetic energy, and work done against resistive forces.

Conservation of energy

Energy is never lost. This is a difficult concept to imagine because we see things using and losing their energy and it took a long time for scientists to realise this. This is the principle of the conservation of energy:

Energy is never destroyed or created, it is only transferred from one form to another. In any energy change the total amount of energy before the change equals the total amount of energy after.

The total amount of energy in a closed system is constant.

This is one of the most important ideas in all of physics and you should be able to remember and apply it to a wide range of situations . There are many quantities in nature that are conserved in nature including mass, charge, strangeness, angular momentum, and baryon number; these last two are covered in another topic.

There are two main forms of energy, kinetic which is essentially the energy of movement, the more kinetic energy an object has the more motion it has and is calculated from the equation:

$$\large E_{k}=\frac{1}{2}mv^{2}$$

Potential energy is the amount of energy stored in an object, generally due to gravity and is worked out from:

$$\large E_{g}=mgh$$

There is also radiant energy, energy contained within electromagnetic radiation and is found from:

$$\large E=hf$$

where f is frequency and h is Plank's constant. Many other actions and instances are related to differenttypes of energy:

  • Thermal or internal energy - Energy related to heat.
  • Electrical energy - Related to the energy in charged objects.
  • Chemical and nuclear energy - Energy associated with chemical or nuclear reactions.
  • Elastic energy - Energy stored when an object is stretched or compressed.

All energy is measured in joules.

One joule is equal to the amount of energy needed to lift a 1N weight through a height of 1m.

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Work

Energy is a measure of an object’s ability to do work, and energy is transferred by either working or heating. They are different in the way that they make objects move.

  • Working makes all the particles in something move in the same way and speed (organised energy). It is an energy transfer when an applied force moves.
  • Heating makes the particles in something move in a very disordered way (disordered energy). It is an energy transfer resulting from a temperature difference.

Work is the product of the force applied and the displacement in the direction of the force.

Work = Force applied × Displacement in the direction of the force

If a force is applied at an angle to an object's movement, the force has a component of Fcosθ in the direction of movement. So the work done is the component of the force acting in the direction of movement × displacement. If the force is acting at 90º to the displacement then the work done on the object will be 0. So a full equation for work can be given as:

$$\large W=F\cos \theta$$

The unit of work is the $\units{Nm}$ or joule(J)

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Stretching Objects

When an object is stretched its tension increases thus the stretching force is increased. The work done to increase the extension (e)of an object to a maximum from 0 is equal to the average force multiplied by e. If the object obeys Hooke's law then the force required is directly proportional to the extension and is parallel to it:

$$\large W=\frac{1}{2}ke^{2}$$

Where k is the spring constant or the force needed to extend the spring.

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Force-Displacement Graphs

Work done can be found by plotting force against distance with force on the x axis. The area under the graph is the work done.

forec displacement graph
Figure 1: A force-displacement graph.

For a simple straight line graph as above in graph A, finding the area and hence the work done is simple. However if the force varies with distance then calculating work done is much harder. Either the graph can be split up into smaller strips and the area of each individual segment can be calculated or you can integrate force with respect to distance. The expression for this can be written as:

$$\large W=\int_{b}^{a}F\delta s$$

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Energy

The amount of energy an object has is related to the amount of work done on it. The more work done then the more energy an object will have. We have already seen the equations for kinetic and gravitational potential energy, but it is worthwhile understanding where they come from.

We can derive the equation for kinetic energy from the amount of work that must be done to accelerate an object to its final velocity from 0. The work done on the object is the force times the distance moved whilst accelerating and can be found from the equations of kinematics:

$$s=\frac{1}{2}\left ( u+v \right )t$$

(from a velocity time graph) and acceleration is $a=\frac{\left ( v-u \right )}{t}$ in both cases u=0 where so substituting the equation for acceleration into $F=ma=\frac{mv}{t}$.

And then the equation for work done W=Fs becomes:

$$\large W=\frac{mv}{t}\times \frac{vt}{2}=\frac{1}{2}mv^{2}$$

As the joule is the unit for energy and work any equation for working out the amount of energy can also be read as the work done on that object.

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Gravitational Potential Energy

This again is calculated from the work equation. To lift an object through a vertical height, work needs to be done against a gravitational field. The force applied is equal to the weight (mg) of the object, and the displacement is the height through which the object is raised.

So as:

$$\large F=mg$$

and:

$$\large s=h$$

We get the familiar:

$$\large E=mgh$$

It is very important to recognise that it is the change in height that is important, and not the absolute height, so we should read this as being the change in gravitational potential energy.

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Energy Changes

As energy is conserved in all interactions we can calculate easily the amount of kinetic energy an object has from the amount of stored potential energy. This can be useful in a number of situations, especially if the acceleration of an object is not constant.

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Power

Energy and can be transferred and work can be done by many means and so sometimes it is more useful to consider how fast energy the is transferred or how quickly the work is done. The rate at which energy is transferred is called the power.

$$\large P=\frac{W}{t}$$

One watt of power transfers one joule of energy per second. The equation above can also be written as, $P=\frac{Fs}{t}$, we can see that it contains a term for velocity, $\frac{s}{t}=v$. This can give us another equation for power:

$$\large P=Fv$$

This is a very important equation and is often overlooked. If we think about someone wading through water, whether they are moving fast or slow, the force from the water is the same, but to move quickly they will need to generate more power than if they were wading slowly.

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