Physics 9702/53 — October/November 2020

Background Concept

The relationship suggested is a power law between extension $x$ and cross-sectional area $A$:

$$x = \frac{mgw^3N A^n}{\gamma\rho}.$$

Here $m$ and $g$ set the applied force $F = mg$, $w$ is the (coil) width, $N$ is the number of turns, $\rho$ is the density of the metal, and $\gamma$ and $n$ are constants to be found from experiment. To test a power law experimentally, it is common to linearise it using logarithms:

$$\ln x = n\ln A + \ln\left(\frac{mgw^3N}{\gamma\rho}\right).$$

This is of the straight-line form $y = mx + c$ if we plot $y = \ln x$ against $x = \ln A$. Then the gradient gives $n$, and the intercept gives the remaining constant combination, allowing $\gamma$ to be calculated.

Understanding the Question

You must design a practical to see how $x$ changes when you change $A$ for different springs, while keeping the other quantities in the formula the same. You are also asked how to use results to determine $\gamma$ and $n$.

So you need:

• A way to make (or obtain) springs of the same material but different wire thickness (different $A$).
• A consistent method to apply the same load $mg$ to every spring.
• A consistent definition and measurement of extension $x$.
• Control of $w$ and $N$ (and ensure the metal is the same so $\rho$ is constant).
• A graph-based analysis to extract $n$ and $\gamma$.

Approach

1. Choose $A$ as the independent variable by using springs made from wires of different diameter.
2. Measure $A$ accurately by measuring wire diameter $d$ with a micrometer and using $A = \pi d^2/4$.
3. Measure extension $x$ as the difference between loaded and unloaded lengths (this cancels any zero error in the ruler position).
4. Repeat measurements and use a range of at least 5 springs (good spread in $A$) to make a reliable graph.
5. Linearise with logs: plot $\ln x$ vs $\ln A$ so the gradient is $n$. Use the intercept plus known constants to calculate $\gamma$.

Step-by-Step Reasoning

(1) Preparing/choosing springs

• Use the same metal for all springs so $\rho$ is constant.
• Keep the number of turns $N$ the same by counting turns during winding.
• Keep the spring width $w$ the same by winding all springs on the same cylindrical former (mandrel) so the coil diameter stays fixed. Check $w$ with calipers.

(2) Measuring the independent variable $A$

• For each spring, measure the wire diameter $d$ using a micrometer.
• Take multiple readings along the wire and in two perpendicular orientations (wires can be slightly non-circular). Average to get a better estimate of $d$.
• Calculate

$$A = \frac{\pi d^2}{4}.$$

This is usually the biggest source of uncertainty because $A$ depends on $d^2$, so careful micrometer use matters.

(3) Measuring the dependent variable $x$

• Hang the spring vertically from a clamp stand.
• Place a metre rule beside it, fixed so it does not move.
• Attach a small horizontal pointer to the bottom end of the spring (or use the bottom coil as an indicator) and read the position using a set square to reduce parallax.
• Measure the unstretched length $L_0$.
• Add the same load of mass $m$ (mass hanger + masses). Wait until it stops oscillating, then measure the loaded length $L$.
• Extension is

$$x = L - L_0.$$

Taking a difference is good practice because it reduces systematic errors from where you chose “zero” on the ruler.

(4) Repeats and quality of data

• Repeat the loading/unloading at least twice (or more) and take the mean extension.
• Use at least 5 (preferably 6–8) different values of $A$ to obtain a reliable best-fit line.

(5) Linearising and extracting $n$
Start from

$$x = \frac{mgw^3N A^n}{\gamma\rho}.$$

Take natural logs:

$$\ln x = \ln\left(\frac{mgw^3N}{\gamma\rho}\right) + n\ln A.$$

So if you plot $\ln x$ against $\ln A$:

• The points should lie close to a straight line if the relationship is correct.
• The gradient of the best-fit line is $n$.

(6) Determining $\gamma$ from the intercept
Let the intercept be $c$ where

$$c = \ln\left(\frac{mgw^3N}{\gamma\rho}\right).$$

Then

$$\frac{mgw^3N}{\gamma\rho} = e^{c}$$

so

$$\gamma = \frac{mgw^3N}{\rho e^{c}}.$$

Use measured $w$, counted $N$, known $m$ and $g$, and the metal density $\rho$ (given or from data book). If using $\log_{10}$ instead, the method is the same but you would use $10^{c}$ instead of $e^{c}$.

Key Takeaways

• A power law $x \propto A^n$ is tested best by a log-log plot.
• $\ln x$ vs $\ln A$ gives a straight line with gradient $n$.
• The y-intercept contains the remaining constants and can be rearranged to find $\gamma$.
• Good planning marks come from clear variables, realistic measurements, control of variables, and a correct graph-based analysis.

Common Mistakes

• Not keeping $m$, $w$ or $N$ constant (then changes in $x$ cannot be attributed only to $A$).
• Measuring only one length and calling it “extension” instead of using $x = L - L_0$.
• Plotting $x$ against $A$ directly and trying to read $n$ without linearising.
• Forgetting that $A$ depends on $d^2$ and not describing how $d$ is measured accurately.
• Not stating how $\gamma$ is obtained from the intercept (many students stop after finding $n$).

Things to Be Careful About

• Elastic limit: loads must be small enough that the spring returns to its original length; otherwise the model is invalid.
• Parallax: always read the scale with a set square/pointer.
• Oscillations: take readings only when the spring is at rest (or gently damp it).
• Consistency of $w$: define clearly what width means (e.g. coil diameter) and keep it the same using the same former.
• Density $\rho$: it must be constant (same metal) and you should state the source (data book or separate measurement).