Energy

Cambridge·Physics·1.7.1·Extended

Energy stores

Energy can be associated with a system because of its state or arrangement. For example, a moving object has energy associated with its motion, while a stretched spring has energy associated with its deformation. These associations are described as energy stores.

A system may have energy in more than one store at the same time. The main stores are:

Kinetic store: energy associated with motion. A moving car has a kinetic store.
Gravitational potential store: energy associated with position in a gravitational field. A raised object has a gravitational potential store.
Elastic, or strain, store: energy associated with deformation, such as the stretching of a spring.
Chemical store: energy associated with the arrangement of particles in substances, such as those in a battery or food.
Nuclear store: energy associated with atomic nuclei.
Electrostatic store: energy associated with the positions of electric charges.
Internal, or thermal, store: energy associated with the microscopic motion of particles within a system and their interactions with one another.

Changes in energy stores

An energy store increases when energy is transferred into it and decreases when energy is transferred out. For example, as an object falls, its gravitational potential store decreases and its kinetic store increases. Energy may also be transferred to other stores, such as the internal store of the surrounding air.

Energy transfer pathways

During a process, energy can move from one store to another. A store describes where energy is associated before or after the process; a transfer pathway describes how the energy moves between stores.

There are four main transfer pathways:

Mechanical work: energy is transferred when a force acts through a distance. For example, lifting an object transfers energy mechanically to its gravitational store.
Electrical work: energy is transferred through an electrical current. For example, a current can transfer energy from a battery’s chemical store to the thermal store of a component.
Heating: energy is transferred because of a temperature difference. For example, when a hot object touches a cooler one, energy is transferred from the hotter object’s thermal store to the cooler object’s thermal store.
Waves: energy is transferred by waves, including electromagnetic waves, sound waves and other waves. For example, light can transfer energy to an object, increasing its thermal store.

In each case, the pathway names the way energy is transferred, not the store in which the energy ends up.

Conservation of energy

Energy can be transferred from one store to another, but it cannot be created or destroyed. This is the conservation of energy: the total energy of a closed system remains constant.

When one store gains energy, that energy must come from elsewhere. For example, if a store gains $10\ \text{J}$ , other stores together must lose $10\ \text{J}$ . Energy may also be transferred to the surroundings. It has not disappeared: it is accounted for in the surroundings rather than in the system being considered.

Conservation in energy flow diagrams

An energy flow diagram uses arrows to show energy entering a process and being transferred out along different pathways. Energy transferred to the surroundings is one possible output. Some energy may instead remain in the process as an increase in a store.

The energy entering is accounted for by the energy leaving and any change in energy stored:

\text{energy input} = \text{energy outputs} + \text{change in stored energy}

For example, if $100\ \text{J}$ enters a process, $60\ \text{J}$ leaves along one pathway, $30\ \text{J}$ is transferred to the surroundings, and $10\ \text{J}$ is added to a store, all $100\ \text{J}$ is accounted for.

A change in stored energy can also be negative: if a store loses energy, the outputs can be greater than the energy entering because some output energy comes from that store.

Where arrow widths are drawn to scale, wider arrows represent greater amounts of energy transferred. Arrows drawn to the same scale can therefore be compared to see the relative amounts transferred along different pathways.

Kinetic energy equation

A moving object has kinetic energy. Its kinetic energy depends on its mass and speed:

E_{\mathrm{k}} = \frac{1}{2}mv^{2}

Here, $E_{\mathrm{k}}$ is kinetic energy in joules (J), $$$m$$$ is mass in kilograms (kg), and $$$v$$$ is speed in metres per second (m/s).

Effect of mass and speed

Kinetic energy is proportional to mass: at the same speed, doubling the mass doubles the kinetic energy.

Kinetic energy is proportional to the square of speed: for the same mass, doubling the speed gives four times the kinetic energy, while tripling the speed gives nine times the kinetic energy.

Finding mass or speed

The equation can be rearranged to find mass or speed when the other two quantities are known:

m = \frac{2E_{\mathrm{k}}}{v^{2}} \qquad \text{and} \qquad v = \sqrt{\frac{2E_{\mathrm{k}}}{m}}

To find speed, take the square root because the original equation contains $v^{2}$ . For example, an object with a mass of $4\ \mathrm{kg}$ and kinetic energy of $200\ \mathrm{J}$ has a speed of $\sqrt{(2 \times 200)/4} = 10\ \mathrm{m/s}$ .

Gravitational potential energy change

An object’s gravitational potential energy changes when its height changes. Raising it increases its gravitational potential energy; lowering it decreases its gravitational potential energy.

Near a planet’s surface, where gravitational field strength is approximately constant, the change is:

\Delta E_{\mathrm{p}} = mg\Delta h

Here:

$\Delta E_{\mathrm{p}}$ is the change in gravitational potential energy, in joules ( $\mathrm{J}$ ).
$$$m$$$ is the object’s mass, in kilograms ( $\mathrm{kg}$ ).
$$$g$$$ is the gravitational field strength, in newtons per kilogram ( $\mathrm{N/kg}$ ).
$\Delta h$ is the vertical change in height, in metres ( $\mathrm{m}$ ).

A rise gives a positive value of $\Delta h$ and therefore a positive energy change. A fall gives a negative value of $\Delta h$ and a negative energy change. Only the vertical height change is used, not the length of the path taken.

For a constant gravitational field strength, the energy change is proportional to mass and vertical height change. For example, doubling the mass doubles the energy change for the same height change; doubling the height change doubles it for the same mass.

If a $2,\mathrm{kg}$ object is raised vertically by $3,\mathrm{m}$ where $g = 10,\mathrm{N/kg}$ , its gravitational potential energy increases by $2 \times 10 \times 3 = 60,\mathrm{J}$ .