Potential energy of a spring
Hooke’s law
A spring is fixed to a wall at one end and to a block at the other. The block rests on a smooth floor. This is the relaxed position, x equals zero.
Pull the block to the right. The spring stretches and pulls back to the left.
Push the block to the left. The spring is compressed and pushes back to the right.
Either way, the spring force always points back towards x equals zero. That is why it is called a restoring force.
For an ideal spring, the force is proportional to the stretch: F equals minus k x. This is Hooke’s law.
k is the spring constant, measured in newtons per metre. A stiff car suspension spring has a large k. A soft pen spring has a small k.
The minus sign says the force is always opposite to the displacement.
Energy stored in a spring
Pull the block slowly from x equals zero out to x m. The spring force grows steadily from zero to minus k x m.
Plot the spring force against x. It is a straight line sloping down through the origin.
The work done by the spring is the area under this line: a triangle with base x m and height k x m. That is minus half k x m squared.
Your pulling force did the opposite: plus half k x m squared. That energy is now stored in the stretched spring.
So the potential energy of a spring is V equals half k x squared. It is the same whether you stretch or compress the spring.
Now let go. At the far ends, the block stops for an instant. All the energy is in the spring.
At the middle, the spring is relaxed and the block is fastest. All the energy is kinetic. Half m v squared equals half k x m squared.
On a smooth floor the energy keeps swapping between the block and the spring, and the total never changes. The spring force is conservative.
Try it: block on a spring
- Press Release and watch the K and V bars swap while E stays fixed.
- The shaded triangle on the graph is the energy stored, ½kx².
- Double A and see the total energy become four times bigger.
Car into a spring
To test crash safety, a 1000 kg car moving at 18 km/h on a smooth road runs into a spring of spring constant 6.25 × 10³ N/m. What is the maximum compression of the spring?
- 0.4 m
- 2 m
- 4 m
- 7.2 m
Answer
B. First convert: 18 km/h = 5 m/s. At maximum compression the car stops, so all its kinetic energy is in the spring: ½mv² = ½kx². Then x = v√(m/k) = 5 × √(1000 / 6250) = 5 × 0.4 = 2 m. Using 18 instead of 5 gives 7.2 m, which is wrong.
Twice the stretch
A spring is stretched by 10 cm and stores 2 J of energy. How much energy does it store when stretched by 20 cm?
- 2 J
- 4 J
- 8 J
- 16 J
Answer
C. Spring energy is V = ½kx², so it depends on x². Doubling the stretch multiplies the energy by 2² = 4: 4 × 2 J = 8 J.
Summary
- Hooke’s law:
F = −kx - Potential energy of a spring:
V = ½kx² - Energy exchange (smooth floor):
½mv² + ½kx² = ½kxₘ² - Fastest speed at x = 0:
vₘ = xₘ√(k/m)
- The spring force is a restoring force: it always points back towards the relaxed position.
- The work done stretching a spring is the triangle under the F–x graph, ½kx². It is stored as potential energy.
- Double the stretch and the stored energy becomes four times bigger.
- The spring force is conservative: on a smooth floor, K + V stays constant as the block oscillates.