Changes in energy: Eₑ = ½ke²

The idea

This equation gives the elastic potential energy stored in a stretched or compressed spring, provided the spring obeys Hooke’s law. Eₑ is the energy in joules (J), k is the spring constant in newtons per metre (N/m), and e is the extension or compression in metres (m). The extension e is the change in length from the natural length, not the total length. Because e is squared, doubling the extension quadruples the stored energy. For example, a spring with k = 200 N/m stretched by e = 0.10 m stores Eₑ = ½ × 200 × (0.10)² = 1.0 J. The factor ½ arises because the force increases linearly from zero to ke, so the average force is ½ke. Always convert centimetres to metres and grams to kilograms where needed before substituting.

What earns the marks

• State that Eₑ is the elastic potential energy stored in the spring, measured in joules (J).
• Identify k as the spring constant in N/m and e as the extension or compression in metres, measured from the natural length.
• Substitute values correctly into Eₑ = ½ke², including squaring the extension before multiplying by ½ and k.
• Convert units where necessary, for example cm to m or N/cm to N/m, before calculating.
• Interpret a calculated energy as the energy transferred to the spring during stretching, or released when it returns to its natural length.

Common mistakes and how to put them right

• Using the total length of the spring instead of the extension.

  Put it right by subtracting the natural length to find e.

• Forgetting to square the extension.

  Put it right by applying the exponent to e before multiplying by ½ and k.

• Mixing units, such as using extension in cm with k in N/m.

  Put it right by converting all lengths to metres and k to N/m.

Exam tips

• Write the equation, then substitute numbers with units before calculating to reduce unit errors.
• Check whether the question gives extension or total length; if total length, subtract the natural length first.
• For 'show that' questions, give the substituted values and the calculated result to at least two significant figures.