Physics 9702/51 — May/June 2010

Background Concept

A power-law relationship has the form

$$d = kv^n$$

where $d$ is the dependent variable (what you measure), $v$ is the independent variable (what you vary), and $k$ and $n$ are constants.

To determine an unknown power $n$, the standard technique is to take logarithms:

$$\log d = \log k + n\log v.$$

This is in the straight-line form $y = c + mx$ if we identify:

• $y \equiv \log d$
• $x \equiv \log v$
• gradient $m \equiv n$
• intercept $c \equiv \log k$

So if the suggested model is correct, a plot of $\log d$ against $\log v$ should be a straight line, and its gradient gives $n$.

Understanding the Question

You are asked to design a workable laboratory experiment that links:

• the speed $v$ of a mass at the moment just before it hits a nail, and
• the depth $d$ that the nail is driven into wood,

and then to process the data to obtain a value for the exponent $n$.

Because this is a planning question, marks come from:

• a clear method (what you do, step-by-step),
• correct and realistic measurements (how you get $v$ and $d$),
• control of other variables (so the test is fair),
• a correct analysis method (log-linearisation and graph),
• sensible safety precautions.

Approach

1. Create impacts at different speeds by varying the release height $h$ of the mass.
2. Measure $v$ as close to the nail as possible (so it represents the speed at impact). A light gate close to the nail is the most direct method.
3. Measure $d$ as a change in nail position (e.g. protruding length before and after). This is often easier and more precise than trying to measure the buried portion directly.
4. Repeat readings and use several values of $v$.
5. Linearise $d = kv^n$ using logs and use a straight-line graph to obtain $n$ from the gradient.

Step-by-Step Reasoning

1) Measuring the speed $v$

• Attach an interrupt card (flag) of known length $L$ to the falling mass.
• Place a light gate a small distance above the nail head.
• As the mass passes the light gate, it blocks the beam for time $t$.
• The speed at that point is

$$v = \frac{L}{t}.$$

Placing the light gate close to the nail makes this measured speed a good approximation to the speed at impact (air resistance and further acceleration over a small gap are small compared with other uncertainties).

2) Varying $v$

• Change the release height $h$ (measured with a metre rule from a fixed reference point).
• Use a wide range of $h$ values (at least 6) to produce a clear range of $v$.
• Release the mass from rest without pushing to keep the method consistent.

(An alternative method, if light gates are unavailable, is to use $v = \sqrt{2gh}$, but then you are not measuring $v$ directly and air resistance may matter; light gates are usually credited as the better plan.)

3) Measuring the penetration depth $d$

• Before the impact, measure the protruding length of the nail above the wood, $x_1$ (e.g. using vernier calipers).
• After one impact, measure the new protruding length $x_2$.
• The depth driven in by that impact is

$$d = x_1 - x_2.$$

This method measures a difference of two lengths, which can be more reliable than trying to judge the depth of the tip inside the wood.

4) Controlling variables
A fair test means only $v$ changes while other factors stay the same.

• Keep $m$ constant: use the same falling mass for all trials.
• Keep nail properties constant: use identical nails (same diameter/shape) and insert to the same initial depth (a marker line on the nail or a physical stop helps).
• Keep wood properties constant: same wood type and thickness; if repeated impacts damage the wood significantly, move to a fresh region or use a new block.
• Keep alignment constant: use a guide tube/rails so the mass hits the nail head centrally every time and does not glance off.
• Keep the number of impacts constant: one impact per data point, otherwise $d$ would depend on the number of strikes as well as speed.

5) Analysis to find $n$
From

$$d = kv^n$$

take logs:

$$\log d = \log k + n\log v.$$

So:

• calculate $\log v$ and $\log d$ for each run,
• plot $\log d$ (y) against $\log v$ (x),
• draw the best-fit straight line,
• the gradient is $n$.

If you include repeats, you can plot means of $d$ at each $v$ and estimate uncertainties (e.g. half-range from repeats for $d$, or instrument precision), and use error bars to judge the reliability of the straight-line fit.

6) Safety

• Clamp the wood securely; keep hands away from the falling mass and nail.
• Wear eye protection and/or use a screen (wood/nail fragments, rebound).
• Ensure the guide prevents lateral motion; provide a safe stop/catcher to prevent the mass from bouncing unpredictably.

Key Takeaways

• To determine an unknown exponent in $d = kv^n$, use a log-log plot: gradient gives $n$.
• Measure impact speed as close as possible to impact (light gate near the nail).
• Measure penetration as a change in protruding length for better precision.
• Good plans explicitly control variables and include repeats.

Common Mistakes

• Trying to find $n$ by plotting $d$ against $v$ directly (does not give $n$ unless $n=1$).
• Not measuring $v$ at (or near) the moment of impact (e.g. using an unrelated speed value).
• Forgetting to keep the initial nail depth the same each run.
• Allowing multiple impacts for one reading of $d$ without stating it and controlling it.
• Vague safety statements ("be careful") without specific hazards and precautions.

Things to Be Careful About

• Define exactly what your measured $d$ represents (total depth or additional depth per impact) and keep it consistent.
• Keep the light gate sufficiently close to the nail so that the measured $v$ is representative of impact speed.
• Use enough different values of $v$ and a wide range to make the log-log line reliable.
• Ensure units and significant figures are sensible; logs require positive values, so avoid any run where $d$ is too small to measure reliably.
• If the wood deforms or cracks, later readings may not be comparable; plan to use fresh positions or fresh samples.