Physics 9702/51 — May/June 2020

Background Concept

A spring obeys Hooke’s law provided it is used within its limit of proportionality:

$$F = kx,$$

where $F$ is the applied force, $x$ is the extension, and $k$ is the spring constant (stiffness). For a given spring, a graph of $F$ against $x$ should be a straight line through the origin; its gradient is $k$.

This planning question also tests how to test a suggested model by changing one variable (here $A$) while controlling others ($\rho, t, N$), and then using a suitable graph to check whether the predicted dependence holds and to determine an unknown constant ($\beta$).

The suggested model is

$$k = \frac{\beta \rho t^4}{A^{3/2}N}.$$

So if $\rho, t, N$ are constant, the prediction is a power law:

$$k \propto A^{-3/2}.$$

Understanding the Question

You are given several springs made from wire of the same thickness $t$, but each spring has a different cross-sectional area $A$ (the area enclosed by the coil when viewed end-on, as in the figure). You must:

1. describe a realistic laboratory method to measure $k$ for each spring,
2. describe how to measure $A$ for each spring,
3. ensure the other variables in the formula ($\rho, t, N$) are controlled/known,
4. choose an analysis that tests whether $k$ varies as $A^{-3/2}$, and
5. explain how to obtain $\beta$ from your results.

Because it asks you to “draw a diagram … showing the arrangement”, an apparatus set-up sketch is required.

Approach

• Make (or use) springs from the same wire material so $\rho$ is constant.
• Keep wire thickness $t$ the same (verify with micrometer) and keep number of turns $N$ the same (count turns).
• Vary only the coil size, so the spring’s cross-sectional area $A$ changes.
• For each spring, measure $k$ by taking several $(F, x)$ readings and finding the gradient of an $F$–$x$ graph.
• Linearise the given relationship by rewriting it in the straight-line form

$$k = C\,A^{-3/2}, \quad \text{where } C = \frac{\beta \rho t^4}{N}.$$

Then a plot of $k$ against $A^{-3/2}$ should be a straight line through the origin, and the gradient gives $C$, from which $\beta$ follows.

Step-by-Step Reasoning

1. Set up to measure extension accurately

   • Hang the spring from a clamp stand.
   • Place a metre rule vertically next to the spring.
   • Use a horizontal set square or a pointer attached to the lower end of the spring so you read the scale without parallax.

2. Measure the geometry that gives $A$

   • The cross-sectional area of the spring coil can be found from its mean diameter $D$:

$$A = \pi\left(\frac{D}{2}\right)^2.$$

• Measure $D$ using vernier calipers. Take several readings in different orientations and average (springs may not be perfectly circular).

1. Measure and control $t$ and $N$

   • Use a micrometer to measure wire thickness $t$ at several points along the wire and average. This both checks that $t$ is the same for all springs and gives a good value for later calculation of $\beta$.
   • Count the turns to find $N$. If the relationship is to be tested fairly, either make all springs with identical $N$ or record $N$ and include it in the analysis (but controlling it is simpler and stronger).

2. Determine $k$ for each spring

   • Add known masses $m$ in steps. For each mass, the applied force is

$$F = mg.$$

• Measure the extension $x$ from the change in the spring length relative to the unloaded length.
• Repeat readings (or do loading and unloading) and average to reduce random error and reduce the effect of hysteresis.
• Plot $F$ vs $x$. The gradient is

$$\text{gradient} = \frac{\Delta F}{\Delta x} = k.$$

Using a graph is better than using a single $F/x$ because it averages over many points and makes it easier to see if Hooke’s law holds (straight line through origin).

1. Test the relationship and find $\beta$
   • With $\rho, t, N$ constant, the model predicts

$$k = \left(\frac{\beta \rho t^4}{N}\right) A^{-3/2}.$$

• For each spring compute $A^{-3/2}$.
• Plot $k$ (y-axis) against $A^{-3/2}$ (x-axis).
  • If the points lie on a straight line through the origin (within uncertainties), the relationship is supported.
• The gradient $G$ satisfies

$$G = \frac{\beta \rho t^4}{N}.$$

• So compute

$$\beta = \frac{GN}{\rho t^4}.$$

Here $\rho$ can be taken from a data book for the metal. If you want to measure it, measure mass with a balance and volume by measuring length of wire used and its cross-sectional area (from $t$) to get density.

1. Uncertainty handling (good practice for full credit)
   • Uncertainty in $x$ is typically dominated by scale reading/parallax; reduce with pointer/set square.
   • Uncertainty in $k$ comes from the gradient uncertainty on the $F$–$x$ graph.
   • Uncertainty in $A$ comes from uncertainty in $D$; since $A \propto D^2$, the percentage uncertainty in $A$ is about twice that in $D$.
   • Taking multiple springs (at least 5–6 different $A$ values) improves the reliability of the straight-line test.

Key Takeaways

• Measure $k$ from the gradient of an $F$–$x$ graph (Hooke’s law).
• Convert a coil diameter measurement into cross-sectional area using $A = \pi(D/2)^2$.
• Control variables that appear in the model ($\rho, t, N$) so only $A$ changes.
• Linearise the model by plotting $k$ against $A^{-3/2}$; gradient gives a constant from which $\beta$ is found.

Common Mistakes

• Using a single reading of $k = F/x$ instead of a graph, losing reliability and making it hard to check Hooke’s law.
• Not stating how $A$ is measured (must relate it to a measurable diameter).
• Forgetting to control or measure $N$ and $t$ even though they appear in the formula.
• Overloading the spring so it plastically deforms; then $F$–$x$ is not linear and $k$ is not well-defined.
• Plotting the wrong linear graph (e.g. $k$ vs $A$ rather than $k$ vs $A^{-3/2}$, or not explaining how $\beta$ comes from the gradient).

Things to Be Careful About

• Definition of $A$: it is the area enclosed by the coil (based on the mean coil diameter), not the cross-sectional area of the wire.
• Mean diameter: for a spring, measure the coil diameter carefully; if you measure outer diameter, you may need to subtract one wire thickness to approximate mean diameter.
• Units: use SI (e.g. $D$ in m so $A$ in $\text{m}^2$; then $A^{-3/2}$ has units $\text{m}^{-3}$).
• Straight line through origin: the model predicts proportionality; a significant intercept suggests systematic error (e.g. zero error in extension).
• Safe working: secure clamp stand, add masses gently, keep clear of falling masses, and stay within elastic limit.