Potential Energy: Complete Guide for UK GCSE and A-Level Physics
Potential energy is stored energy that an object has due to its position or configuration. In UK Physics, students encounter two main forms: gravitational potential energy (GPE), stored by objects at height, and elastic potential energy, stored in stretched or compressed materials. This calculator covers all three essential calculations for GCSE and A-Level examinations.
Gravitational Potential Energy: Ep = mgh
Gravitational potential energy is the energy stored by an object due to its height above a reference point. The formula is:
Ep = m × g × h
- Ep = gravitational potential energy (joules, J)
- m = mass (kilograms, kg)
- g = gravitational field strength (9.81 m/s² on Earth's surface)
- h = height above the reference point (metres, m)
For GCSE, g is often rounded to 9.8 or 10 m/s² to simplify calculations. The accepted value is 9.81 m/s² and this should be used for A-Level and more precise calculations.
Worked Example: Ball on a Shelf
A 2 kg ball is placed on a shelf 3 m above the floor. Calculate its GPE relative to the floor.
- Ep = m × g × h = 2 × 9.81 × 3 = 58.86 J
If the ball falls, this 58.86 J of GPE converts to kinetic energy. Assuming no air resistance: KE = ½mv² = 58.86 J, giving v = √(2 × 58.86 / 2) = √58.86 ≈ 7.67 m/s when it hits the floor.
Elastic Potential Energy: Ee = ½kx²
Elastic potential energy (also called elastic strain energy) is stored in materials that are stretched or compressed within their elastic limit. The formula is:
Ee = ½ × k × x²
- Ee = elastic potential energy (joules, J)
- k = spring constant (N/m) - a measure of stiffness
- x = extension or compression (metres, m)
This formula is derived from Hooke's Law (F = kx) and the work done equation. The elastic potential energy equals the area under a force-extension graph: Ee = ½Fx = ½kx².
Energy Conservation and Conversion
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred between stores. This is a central idea in GCSE Physics and underpins all potential energy calculations:
- Falling object: GPE → Kinetic energy (Ep = ½mv² when it reaches the ground)
- Pendulum: GPE at top → KE at bottom → GPE at top (continuously cycling)
- Roller coaster: GPE at peak → KE at bottom → GPE at next peak (minus friction losses)
- Compressed spring: Ee → KE when released (e.g. in a spring-loaded toy)
- Stretched bungee cord: KE (falling) → Ee (at maximum stretch) → KE again
Gravitational Field Strength on Other Planets
The value of g varies on different celestial bodies, affecting the GPE of objects at the same height and mass:
- Earth: g = 9.81 m/s²
- Moon: g = 1.62 m/s² (about 1/6 of Earth's)
- Mars: g = 3.72 m/s²
- Jupiter: g = 24.79 m/s²
- Mercury: g = 3.7 m/s²
Our calculator allows you to enter any value of g, making it useful for multi-planetary physics problems at A-Level.
Spring Constant and Hooke's Law
The spring constant (k) measures the stiffness of a spring or elastic material. A stiffer spring has a higher k value and stores more energy for the same extension. Typical spring constants:
- Slinky toy: approximately 1 N/m
- Typical lab spring: 10-100 N/m
- Car suspension spring: 10,000-30,000 N/m
- Stiff industrial spring: 100,000+ N/m
The spring constant is found experimentally by measuring force applied against extension and calculating the gradient of the force-extension graph: k = F/x (within the elastic limit).
Finding Height from GPE
Rearranging Ep = mgh to find height: h = Ep / (m × g). This is useful in examination questions where you are given the energy of an object and need to find the height it was dropped from, or the height it could reach. For example, if a 3 kg object has 147.15 J of GPE, its height is h = 147.15 / (3 × 9.81) = 5 m.
Exam Tips for Potential Energy Questions
- Always identify your reference point (usually the ground or the lowest point in the problem).
- Check units: mass must be in kg, height in m, and g in m/s².
- For elastic PE, extension x must be in metres - convert cm by dividing by 100.
- In energy conservation problems, state which energy stores are involved at each point.
- At A-Level, note that GPE can be negative (below the reference level).