Maths › Further Mechanics 1 › Elastic potential energy
Elastic potential energy
A stretched string holds energy, and the amount is the area under the Hooke's law line. Adding that term to the energy equation extends the work-energy principle to every spring problem on the paper.
Builds on Hooke's law and elastic strings and Work, energy and power.
IN THIS TOPIC
- Derive and use the formula for the energy stored in a stretched string.
- Include elastic energy in a work-energy equation.
- Solve problems where a particle is projected by a spring or oscillates on a string.
COMMON MISCONCEPTION
The elastic energy stored is the tension times the extension.
The area under the line
The tension grows from zero to λx/l as the string stretches, so the work done in stretching it is the area under the straight Hooke's law graph. That area is a triangle of base x and height λx/l.
The booklet prints nothing on elastic strings, so this and Hooke's law are both memorise items. The factor of a half is essential. Tension times extension would be the work done by a constant force of the final size, and the force was never constant. It started at nothing. Notice also the x², so doubling the stretch stores four times as much.
WORKED EXAMPLE
Energy in a stretched string
An elastic string of natural length 1.5 m and modulus 60 N is stretched to 2 m. Find the energy stored.
Extension = 0.5 m, and the tension at that extension is 20 N.
EPE = 60 × 0.25/(2 × 1.5) = 15/3 = 5 J.
Checking by area: ½ × 0.5 × 20 = 5 J, which is the same triangle read a different way.
Adding it to the energy equation
With elastic energy in hand, the work-energy principle covers everything on this paper. Kinetic energy, gravitational potential energy, elastic energy, and work done against friction all go into one account. Write down the total at one instant and the total at another, then set the difference equal to whatever friction has taken.
The commonest mistakes here are geometric. Students use the total length where the extension is wanted, forget that a string goes slack once its ends are closer than the natural length, or measure heights from different levels in the two totals. Draw the two instants and label a single zero level for height.
GUIDED PRACTICE
A spring launcher
A particle of mass 0.5 kg is held against a spring of natural length 0.8 m and modulus 40 N, compressed by 0.2 m, on a smooth horizontal table. Find its speed when the spring reaches its natural length.
Show the working
Energy stored = 40 × 0.2²/(2 × 0.8) = 1.6/1.6 = 1 J.
The table is smooth and horizontal, so no energy is lost and no height changes.
All 1 J becomes kinetic: ½(0.5)v² = 1, so v² = 4 and v = 2 m/s.
Doubling the compression would store 4 J and double the speed, since the energy goes as the square of the compression and the speed as the square root of the energy.
ASSESSMENT FOCUS
- Include the factor of a half. Tension times extension is twice the energy.
- Use the extension, not the length, and square it.
- Choose one zero level for height and use it in both energy totals.
- Check whether a string is slack at either instant, since a slack string stores nothing.
CHECK YOURSELF
An elastic string of natural length 2 m and modulus 49 N is stretched by 0.5 m. Find the energy stored.
Show a hint
Square the extension and halve.
Show the answer
EPE = 49 × 0.25/(2 × 2) = 12.25/4 = 3.0625 J, about 3.06 J.
The energy stored is λx²/(2l), the area of the triangle under the Hooke's law line, so it grows with the square of the extension.
Add it to kinetic and gravitational potential energy, and set the total change equal to any work done against friction.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the elastic potential energy questions page.
CHECK YOUR PROGRESS
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- Derive and use the formula for the energy stored in a stretched string.
- Include elastic energy in a work-energy equation.
- Solve problems where a particle is projected by a spring or oscillates on a string.
Open the full revision checklist to see every objective in the course in one place.