Elastic strain energyEdexcel A-Level Physics: Revision notes

To stretch a spring or wire, a force must do work. If the force were constant, work = force × distance. A stretching force is not constant, because it rises from zero as the extension increases. Within the elastic limit this work is stored in the material as elastic strain energy and is recovered when the force is removed.

The work done equals the area under the force–extension graph.

For a material obeying Hooke's law, the graph is a straight line through the origin, so the area is a triangle with base x and height F:

$E_{el} = \tfrac{1}{2}Fx$

The average force during the stretch is F/2, which is why the energy is half of F × x. Since F = kx, this is also ½kx². Doubling the extension doubles both F and x, so the energy rises four times.

Worked example: k = 250 N m⁻¹, x = 0.060 m. F = 15 N, so E = ½ × 15 × 0.060 = 0.45 J. At x = 0.120 m, E = ½ × 30 × 0.120 = 1.8 J.

When the graph is not a straight line, ½Fx does not apply, so the energy must be found from the area under the curve. Two methods:

• Counting squares: find the energy represented by one square, and count whole and part squares.
• Trapezia (or strips): divide the area into strips of equal width h. Each trapezium has area ½(F₁ + F₂)h; add them up.

Worked example: a polymer gives 0, 3.0, 5.0 N at extensions 0, 2.0, 4.0 mm. Trapezia: ½(0 + 3.0) × 2.0 + ½(3.0 + 5.0) × 2.0 = 3.0 + 8.0 = 11 N mm = 1.1 × 10⁻² J. Convert N mm to J by multiplying by 10⁻³.

For a curve whose gradient decreases, the chords lie below the curve, so trapezia underestimate the area.

Elastic strain energy can be transferred to other forms when the force is removed. For a spring launching a trolley on a frictionless track, all the stored energy can become kinetic energy:

$\tfrac{1}{2}Fx = \tfrac{1}{2}mv^2$

In real systems some energy is transferred to thermal energy and sound, or to the kinetic energy of other parts, so the speed is less than this ideal value. If the energy is multiplied by a factor n, the speed is multiplied by √n, since v ∝ √E.