Physics 9702/51 — May/June 2019

Background Concept

A cantilever is a beam fixed at one end and free at the other. When a load is applied at the free end, the beam bends and the end drops by a vertical deflection $s$. For small deflections (elastic behaviour), the deflection is related to the load and the geometry of the beam.

The question gives the suggested relationship

$$E = \frac{4MgL^3}{bst^3}$$

where:

• $E$ is Young modulus (a material property)
• $M$ is the load mass, so the force is $Mg$
• $L$ is the unsupported length (overhang)
• $b$ is width and $t$ is thickness of the strip
• $s$ is the deflection at the free end

To test the relationship between $s$ and $L$, we want to see whether $s$ varies as a power of $L$ (here, proportional to $L^3$ when other quantities are fixed). To determine $E$, we use the gradient of a suitable straight-line graph.

Understanding the Question

You must design an experiment (Paper 5 planning style):

• Choose what you vary and what you measure.
• Explain how you will measure $L$, $s$, $b$, $t$, and keep $M$ constant.
• Explain how you will process data to check the suggested dependence of $s$ on $L$.
• Explain how to obtain a numerical value of $E$ from your results.
• Include a labelled diagram of the apparatus and safety points.

The key idea is that you can control $M$, $b$, $t$ by keeping the same strip and same load, and vary only $L$. Then measure $s$ for each $L$.

Approach

1. Choose variables: vary $L$ (independent), measure $s$ (dependent).
2. Control variables: keep $M$ constant; keep the strip the same (so $b,t$ constant); always place the load at the same point (the free end).
3. Measurement strategy:
   • Measure $L$ from the support edge to the free end.
   • Measure $s$ as the change in vertical position of the free end when loaded compared with unloaded to reduce zero-offset issues.
   • Measure $b$ and $t$ carefully (especially $t$ because it is cubed).
4. Linearise the relationship so that a straight-line graph can be used:
   • Rearrange to $s = k L^3$ where $k = \frac{4Mg}{Ebt^3}$.
   • Plot $s$ against $L^3$; gradient gives $k$, then solve for $E$.

Step-by-Step Reasoning

1) Setting up the cantilever

Clamp the strip to the bench so one end is fixed and a length $L$ overhangs. The support must not slip during loading; otherwise the effective $L$ changes and ruins the data.

2) Measuring $b$ and $t$

• Use vernier callipers for $b$.
• Use a micrometer screw gauge for $t$.
• Take readings at several points along the strip and average because wood thickness may vary.

This matters because $t$ is in $t^3$, so a small percentage error in $t$ produces roughly three times that percentage error in $t^3$.

3) Measuring $L$

For each trial:

• Slide the strip so the overhang from the support edge to the free end is the chosen $L$.
• Measure $L$ with a metre rule; read at eye level to reduce parallax.

4) Measuring $s$ reliably

Directly measuring a small deflection with a ruler can be difficult. Better options:

• Dial gauge with its tip under the free end.
• Travelling microscope viewing a pointer attached at the free end.
• A ruler fixed vertically with a set square or pointer to align with the end.

Crucially, record an unloaded reading and a loaded reading, then

$$s = \text{(loaded position)} - \text{(unloaded position)}$$

This removes systematic offsets (e.g. the strip not being perfectly horizontal initially).

Repeat $s$ readings for each $L$ and average to reduce random uncertainty.

5) Choosing the data range

Use at least 6 values of $L$ spanning as wide a range as possible while staying in the elastic region (no permanent bend). A wide range gives a clearer trend and a more reliable gradient.

6) Linearising and extracting $E$

Start from the given expression and rearrange for $s$:

$$E = \frac{4MgL^3}{bst^3} \quad \Rightarrow \quad s = \frac{4Mg}{Ebt^3} L^3$$

Let

$$y = s, \quad x = L^3$$

Then

$$y = mx$$

with

$$m = \frac{4Mg}{Ebt^3}$$

So:

• Plot $s$ on the y-axis against $L^3$ on the x-axis.
• Draw a best-fit straight line.
• Determine the gradient $m$ using a large triangle.

Finally, rearrange for $E$:

$$E = \frac{4Mg}{m b t^3}$$

(Use consistent SI units: $L$ and $s$ in $\text{m}$, $b$ and $t$ in $\text{m}$, $M$ in $\text{kg}$.)

7) Considering uncertainty (what you would discuss in evaluation)

• The main uncertainty often comes from measuring $s$ and from $t$.
• If you draw a worst acceptable line as well as the best-fit line, you can estimate uncertainty in gradient $\Delta m$ and hence uncertainty in $E$.

Key Takeaways

• A planning question must clearly specify variables, apparatus, measurements, and how the data will be analysed.
• Testing $s \propto L^3$ is best done by plotting $s$ against $L^3$ for a straight-line check.
• Once linearised, the gradient links directly to $E$.
• Control of variables (fixed $M$, same strip, load always at the free end) is essential for a valid test.

Common Mistakes

• Varying $M$ as well as $L$: then changes in $s$ cannot be attributed solely to $L$.
• Not stating how $s$ is measured (or not using the change from unloaded to loaded).
• Plotting $s$ against $L$ instead of $L^3$, which will not produce a straight line.
• Forgetting to measure $b$ and $t$ (or using a ruler for $t$), giving very large uncertainty in $E$.
• Not keeping the load position fixed: moving the load changes the bending moment.

Things to Be Careful About

• Keep units consistent; use metres for all lengths when calculating $E$.
• Ensure the strip does not slip in the clamp; any slip changes $L$.
• Avoid large deflections: the formula assumes small deflection and elastic behaviour.
• Measure $t$ carefully; because of $t^3$, even a small reading error strongly affects $E$.
• When finding gradient, use a large triangle and compute $\Delta y / \Delta x$ (not $x/y$).