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Spec links: 4.5.3 Forces and elasticity (including Required practical 6: force and extension) · 4.1.1.2 Changes in energy (elastic potential energy, gravitational potential energy) · Reading time: about 4 minutes

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A jump into history

On 1 April 1979, a group of students from Oxford climbed onto the Clifton Suspension Bridge in Bristol. They tied stretchy cords to their ankles and jumped. Many people thought it was an April Fool’s joke. It wasn’t. Modern bungee jumping had begun. Today, people pay to leap from bridges, cranes and towers all over the world.

The jumper’s life depends on one thing: the cord must stretch, stop them safely, and spring back. To understand how, we need to look at how forces change the shape of objects.

Stretching and springing back

To stretch, squash or bend something, you need more than one force. If you pull on a cord, something must hold the other end. A bungee cord is fixed to the bridge at the top and pulled by the jumper at the bottom.

If an object goes back to its original shape and length when the forces are removed, it has been elastically deformed. If it stays stretched or bent, it has been inelastically deformed. A good bungee cord must deform elastically, jump after jump.

Hooke’s law

For a spring, the extension is directly proportional to the force applied. Double the force and you double the extension. This is often called Hooke’s law. We write it as F = k e, where F is force in newtons (N), e is extension in metres (m) and k is the spring constant in newtons per metre (N/m). A stiff spring has a large spring constant.

This only works up to a point. Beyond the limit of proportionality, the extension is no longer proportional to the force. On a force–extension graph, the straight line starts to curve. Stretch even further and the object may not return to its original length at all.

Real bungee cords are made of many rubber strands. They only roughly follow Hooke’s law, but it is a useful model.

Testing it in the lab

In the required practical, you hang a spring from a clamp stand next to a metre rule. You add masses one at a time and measure the new length each time. Extension is the new length minus the original length. Each 100 g mass adds about 1 N of weight. You then plot a graph of force against extension. The straight part of the line shows the spring obeying Hooke’s law, and its gradient gives the spring constant.

Where does the energy go?

At the top of the bridge, the jumper has energy in their gravitational potential energy store. Its size is given by E_p = m g h. A 70 kg jumper falling 35 m loses E_p = 70 × 9.8 × 35 = 24 010 J of gravitational potential energy.

As the cord stretches, energy is transferred to its elastic potential energy store. Provided the limit of proportionality is not passed, E_e = ½ k e². Imagine a cord with a spring constant of 200 N/m stretched by 15 m. The force it exerts is F = 200 × 15 = 3000 N. The energy stored is E_e = ½ × 200 × 15² = ½ × 200 × 225 = 22 500 J. That is most of the energy the jumper lost. The rest is in their kinetic energy store or has been dissipated by air resistance.

Then the cord pulls back. Energy moves back into the jumper’s kinetic and gravitational stores, and they bounce up again. Each bounce is a little lower, until they hang still and wait to be lowered to safety.

Key words

• Elastic deformation: a change of shape where the object returns to its original shape and length after the forces are removed.
• Inelastic deformation: a change of shape where the object does not return to its original shape and length after the forces are removed.
• Extension: the increase in length of an object when it is stretched (new length minus original length).
• Spring constant (k): the force needed per metre of extension, in N/m; a measure of stiffness.
• Limit of proportionality: the point beyond which the extension is no longer directly proportional to the force applied.
• Elastic potential energy: the energy stored in a stretched or compressed elastic object.

Questions

1. In which year did modern bungee jumping begin, according to the article? [1 mark]
2. Why is more than one force needed to stretch a bungee cord? [1 mark]
3. Give the unit of the spring constant. [1 mark]
4. Explain the difference between elastic deformation and inelastic deformation. Use a bungee cord as an example. [2 marks]
5. A pupil plots a force–extension graph for a spring. The line is straight at first, then curves. Explain what the graph shows. [3 marks]
6. A spring has a spring constant of 40 N/m. A force of 6.0 N is applied to it.
   a) Calculate the extension of the spring in metres. [2 marks]
   b) Calculate the elastic potential energy stored in the spring. Give the unit. [2 marks]
7. A bungee jumper of mass 60 kg falls 25 m before coming to a stop. Gravitational field strength = 9.8 N/kg. Calculate the gravitational potential energy lost by the jumper. Give the unit. [3 marks]
8. Describe how a pupil could carry out an investigation to find the relationship between the force on a spring and its extension. Include how they would make the results accurate and safe. [6 marks]

Show answers

1. 1979 (1)
2. Something must hold / pull the other end of the cord (otherwise a single force would just move the whole cord) (1)
3. N/m / newtons per metre (1)
4. Elastic: the cord returns to its original length when the force is removed (1); inelastic: the cord stays stretched / does not return to its original length (1)
5. Straight part: extension is directly proportional to force / the spring obeys Hooke’s law (1); the curve starts at the limit of proportionality (1); after this, extension is no longer proportional to force (1)
6. a) e = F ÷ k = 6.0 ÷ 40 (1); e = 0.15 m (1)
   b) E_e = ½ × 40 × 0.15² (1); E_e = 0.45 J (1) (allow error carried forward from part a)
7. E_p = m g h = 60 × 9.8 × 25 (1); E_p = 14 700 (1); J (1)
8. Indicative content: hang the spring from a clamp stand with a metre rule clamped vertically beside it; measure the original (unstretched) length of the spring; add a known mass / weight, e.g. 100 g (about 1 N), and measure the new length; extension = new length − original length; repeat, adding one mass at a time, for at least five or six forces; read the ruler at eye level to avoid parallax error; use a pointer fixed to the bottom of the spring; check the spring returns to its original length (has not passed the elastic limit); plot force against extension; a straight line through the origin shows extension is directly proportional to force; safety: clamp the stand to the bench or use a counterweight, and wear eye protection in case the spring snaps or flies off.
   Level 3 (5–6 marks): a clear, logically ordered method that would give valid results, including how extension is calculated, steps to improve accuracy, a safety point and how the relationship is found from a graph; Level 2 (3–4 marks): a method that could be followed with some detail of measurements, may lack accuracy or safety points; Level 1 (1–2 marks): simple statements about adding masses or measuring length, not in a logical order.

More reading: See all 30 GCSE Physics comprehension articles