Physics 9702/53 — May/June 2019

Background Concept

A cantilever is a beam that is fixed at one end and free at the other. When a load is applied at the free end, the beam bends and the free end moves down by a deflection $s$.

For a given material and cross-section, the stiffness depends strongly on the thickness: for a rectangular cross-section, the bending stiffness contains a $t^3$ factor, which is why careful measurement of $t$ is important.

The question provides a suggested relationship

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

where:

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

To test how $s$ depends on $L$, we rearrange the given equation so that the dependent variable ($s$) is on one side and the independent variable ($L$) is in a form that should give a straight-line graph.

Understanding the Question

You must plan an experiment where you:

1. change $L$ deliberately,
2. measure the resulting deflection $s$ caused by a fixed load $M$ at the end,
3. keep other factors ($M$, $b$, $t$, the material, and the loading position) constant,
4. analyse the data with an appropriate graph to check the relationship,
5. use the graph to calculate $E$.

Because the relationship contains $L^3$, a direct plot of $s$ against $L$ will be curved; you should instead linearise by using $L^3$.

Approach

• Choose $L$ as the independent variable and vary it over a sensible range (enough to see a clear change in $s$ but not so large that the wood is permanently deformed).
• For each $L$, measure the deflection at the same point (the free end $P$) with and without the load, and take the difference to reduce zero error.
• Compute $L^3$ and plot $s$ vs $L^3$.
• Use the straight-line gradient to obtain $E$ using algebra.

Step-by-Step Reasoning

1. Set up the cantilever

   • Clamp the strip to the bench so one end is fixed.
   • Ensure the bench edge defines the start of the overhang; measure from this edge to the free end $P$.

2. Choose variables and controls

   • Vary only $L$.
   • Keep $M$ fixed so the applied force $Mg$ is constant.
   • Use the same wooden strip so the material and cross-section are unchanged.
   • Always apply the load at $P$ (if the load position changes, the bending moment changes and the formula would no longer apply).

3. Measure the geometry accurately

   • Measure $t$ with a micrometer (better resolution than a ruler). Since the equation uses $t^3$, a small error in $t$ causes a much larger percentage error in $t^3$.
   • Measure $b$ with vernier calipers.
   • Take multiple readings along the strip and average because wood may not be perfectly uniform.

4. Measure deflection $s$ with reduced systematic error

   • Put a ruler vertically near the free end and a pointer at $P$.
   • Record an initial reading with no mass.
   • Hang the mass $M$ at $P$, wait for oscillations to stop, record the new reading.
   • Deflection is the difference: $s = y_{\text{loaded}} - y_{\text{unloaded}}$.
   • Repeat and average to reduce random error.

5. Collect a suitable dataset

   • Use at least 6 values of $L$ spread over a range.
   • Avoid so large a load/overhang that the strip yields (permanent bend) because the given relationship assumes elastic behaviour.

6. Linearise and test the relationship
   Starting from the given equation, rearrange to make $s$ the subject:

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

This has the form $s = k L^3$ where $k$ should be constant if the model is correct.

1. Graph and determine $E$
   • Calculate $L^3$ for each $L$.
   • Plot $s$ (y-axis) against $L^3$ (x-axis).
   • If the points lie on a straight line through the origin (within uncertainty), that supports $s \propto L^3$.
   • Find gradient $m = \Delta s / \Delta (L^3)$.

From the linear form,

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

so

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

1. Uncertainty ideas (what you would do in practice)
   • Repeat $s$ readings and use spread to estimate uncertainty.
   • Uncertainty in $L$ comes from reading the metre rule and defining the bench edge and point $P$.
   • For the graph, draw a best-fit line and (if required) a worst acceptable line to estimate uncertainty in $m$, then propagate to $E$.

Key Takeaways

• A planning question is marked for: clear variables, workable procedure, good measurement technique, control of variables, correct linearising graph, and a clear route to the required constant ($E$).
• Always rearrange the given relationship into a straight-line form and state what the gradient represents.
• Measure quantities that appear with powers (like $t^3$) with high precision.

Common Mistakes

• Plotting $s$ against $L$ instead of $L^3$, giving a curve and no straightforward way to extract $E$.
• Not keeping the load position fixed at the free end (changes the bending behaviour).
• Measuring $s$ from a single loaded reading without an unloaded reference (zero/parallax errors become systematic).
• Using a ruler to measure $t$ (too low precision; error in $t^3$ becomes large).
• Allowing the strip to slip in the clamp or exceed the elastic limit, invalidating the model.

Things to Be Careful About

• Define $L$ consistently: from the bench edge (clamp line) to the point of loading $P$.
• Wait for oscillations of the hanging mass to stop before taking readings.
• Ensure the vertical scale is fixed and read at eye level (use a set square) to minimise parallax.
• Use consistent units (e.g. convert $b$, $t$, $L$, $s$ to metres before calculating $E$).
• Because $t$ is cubed, take multiple micrometer readings and check for zero error on the micrometer.