Work, Energy and Power
Energy Stores and the Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed — it can only be transferred between stores. In any closed system, total energy before a process equals total energy after. Energy that seems to ‘disappear’ has been dissipated — spread into less useful thermal energy in the surroundings.
Energy can be stored in different ways: kinetic (moving objects), gravitational potential (objects at height), elastic potential (stretched or compressed objects), chemical (fuels, food, batteries), thermal (hot objects), magnetic, electrostatic and nuclear. Energy transfers between these stores via heating, radiation, electrical current or mechanical work.
Energy cannot be created or destroyed. It transfers between stores — kinetic, gravitational potential, elastic potential, chemical, thermal, magnetic, electrostatic, nuclear. Dissipated energy is wasted as thermal energy in the surroundings.
Work Done
When a force moves an object through a distance, work is done. Work done equals the amount of energy transferred. The equation is:
$$W = F \times d$$
where \(W\) is work done in joules (J), \(F\) is the applied force in newtons (N) and \(d\) is the distance moved in the direction of the force in metres (m). One joule of work is done when a force of 1 N moves an object through 1 m.
Work done on an object equals the energy transferred to it. When you push a box across a rough floor, work is done against friction and energy is transferred to the thermal stores of the box and floor. This is why friction causes objects to warm up during sliding — kinetic energy is continuously converted to thermal energy in the surroundings.
Gravitational Potential Energy
When an object is raised against gravity, energy is stored in its gravitational potential energy (GPE) store:
$$GPE = m \times g \times h$$
where \(m\) is mass in kg, \(g\) is the gravitational field strength (10 N/kg for OCR calculations) and \(h\) is the height gained in metres. For example, lifting a 4 kg book to a shelf 1.5 m high: \(GPE = 4 \times 10 \times 1.5 = 60\) J.
Kinetic Energy
Any moving object has kinetic energy (KE):
$$KE = \frac{1}{2} \times m \times v^2$$
where \(m\) is mass in kg and \(v\) is speed in m/s. Notice that KE depends on \(v^2\) — doubling the speed quadruples the kinetic energy. This explains why stopping distances increase so sharply at higher speeds: a car at 30 m/s has four times the KE of one at 15 m/s and requires four times the braking distance.
A rollercoaster car of mass 500 kg is raised to a height of 40 m before being released.
Step 1 — GPE at the top: \(GPE = 500 \times 10 \times 40 = 200{,}000\) J = 200 kJ.
Step 2 — KE at the bottom (assuming no friction): by conservation of energy, all GPE converts to KE. \(KE = 200{,}000\) J.
Step 3 — Speed at the bottom: \(200{,}000 = \frac{1}{2} \times 500 \times v^2\) → \(v^2 = \frac{200{,}000 \times 2}{500} = 800\) → \(v = \sqrt{800} \approx 28.3\) m/s.
In practice, friction and air resistance would dissipate some energy as thermal energy, so the actual speed at the bottom would be lower than 28 m/s.
Work done: \(W = F \times d\) (joules). Gravitational PE: \(GPE = m \times g \times h\) (joules). Kinetic energy: \(KE = \frac{1}{2}mv^2\) (joules). Doubling speed quadruples KE.
Power
Power measures how quickly energy is transferred. The equations are:
$$P = \frac{E}{t} \quad \text{and} \quad P = \frac{W}{t}$$
where \(P\) is power in watts (W), \(E\) or \(W\) is energy transferred or work done in joules (J), and \(t\) is time in seconds (s). One watt equals one joule per second. A device with a higher power rating transfers the same amount of energy in less time — a 2000 W kettle boils water in half the time of a 1000 W kettle.
No energy transfer is perfectly efficient — some energy is always dissipated as thermal energy. Efficiency is calculated as:
$$\text{Efficiency} = \frac{\text{useful output energy}}{\text{total input energy}}$$
A result of 0.7 means 70% efficient (30% wasted). For example, a motor that receives 500 J of electrical energy and does 350 J of useful mechanical work: efficiency = \(\frac{350}{500} = 0.70\) = 70%. The remaining 150 J is dissipated as heat and sound.
Engineers reduce waste energy through lubrication (less friction), streamlining (less air resistance), thermal insulation (less heat loss) and using more efficient technologies such as LED lighting instead of traditional incandescent bulbs.
Elastic potential energy is stored when a spring or elastic material is deformed:
$$EPE = \frac{1}{2} \times k \times e^2$$
where \(k\) is the spring constant in N/m and \(e\) is the extension in metres. Like KE, EPE depends on the square of the deformation — doubling the extension stores four times the energy. This equation applies only within the limit of proportionality, where force and extension are directly proportional (Hooke’s Law).