Chapter 5: Review

The chapter problem, solved

The spring that stopped behaving. You hang weights on a spring and measure how far it stretches. Up to a point each extra newton stretches it by the same amount; then it starts stretching further for the same extra force.

What the graph shows. Force plotted against extension rises as a straight line through the origin while the spring obeys Hooke's law. The gradient of that line is the spring constant:

An experiment giving 2.0 N at 0.05 m and 6.0 N at 0.15 m has

The line stops being straight at the limit of proportionality. Past that point the extension is no longer proportional to the force, so the graph curves upwards: each extra newton gives a larger extension than the one before. The spring's material is being stretched beyond what it can spring back from.

Stretch it further and it reaches the elastic limit, past which it does not return to its natural length when the load is removed. The limit of proportionality comes first; the elastic limit is the point of no return.

Why the energy formula stops working there too. The energy stored is the area under the graph. Below the limit that area is a triangle, which is where $\tfrac{1}{2}kx^2$ comes from. Past the limit the shape is no longer a triangle, so the formula no longer gives the energy — you would have to find the area some other way.

Mixed practice — choose the method first

Work through these without notes. For each one, decide which relationship applies and say why. One of them is deliberately a question you should refuse to answer with the formula you have been using.

1. A spring of constant 60 N/m is stretched by 0.15 m. Calculate the force and the energy stored.
2. A force of 9.0 N stretches a spring by 0.030 m. Calculate its spring constant, then the extension a force of 15 N would give.
3. Two springs of 80 N/m and 20 N/m are joined in series, and then in parallel. Calculate both combined spring constants. Check each against the size of the two.
4. A spring's graph is straight up to 0.20 m and curves afterwards. A force of 20 N gives an extension of 0.25 m. Explain why you cannot use $F = kx$ here, and say what you would do instead.
5. A 250 g mass hangs on a spring and stretches it by 0.049 m. Calculate the spring constant. Take $g = 9.8\,\text{N/kg}$.
6. An extension is measured as 0.020 m on a metre stick read to the nearest millimetre. Calculate the percentage uncertainty, and compare it with the same uncertainty on a 0.100 m extension.
7. A spring stores 3.0 J when stretched by 0.10 m. Calculate the energy stored at 0.30 m.
8. A student plots force against total length instead of extension. Describe what is wrong with the graph and how it would show up in the shape of the line.
9. Two identical springs in series hold up a 2.0 kg mass. Each has a spring constant of 400 N/m. Calculate the extension of the combination.
10. Explain, in terms of what stays the same, why springs in parallel have a larger combined spring constant than either spring on its own.

What you can now do

• state Hooke's law and use $F = kx$ in any direction, with units;
• tell an extension from a length, and say why the difference matters;
• find the combined spring constant of springs in series and in parallel, and check the answer against the size of the individual constants;
• calculate elastic potential energy, and explain why doubling the extension multiplies it by four;
• find a spring constant from the gradient of a force-extension graph, and say which part of the graph to use;
• describe the sources of error in the investigation and calculate the percentage uncertainty in a reading.

Thinking back

Look at the object you chose at the start of the chapter — a bungee cord, a mattress, a pen clip, a shoe sole.

You can now say something quantitative about it. Estimate the force it carries and the distance it moves, and work out a spring constant in N/m. Then answer the question you started with: what happens if it is stretched too far?

If your estimate is rough, that is worth writing down too. Saying how rough an estimate is — the uncertainty — is part of the physics, not an apology for it. It is also exactly what the Physics In Practice Investigation asks you to do.