Physics 9702/52 — May/June 2019

Background Concept

A block topples when the line of action of the resultant downward force passes through the edge (pivot line) about which it tips. In practice, the “just topples” condition is identified by the instant one edge just begins to lift.

The question provides a suggested relationship linking the load mass $m$ and the toppling angle $\theta$:

$$\alpha Vw = 2mL\cos\theta - mw$$

Here $\alpha$ is an unknown constant. $V$ and $w$ depend only on the block dimensions, and $L$ is a fixed distance set by where you attach the load.

To test a relationship experimentally, we vary one variable (here $m$), measure the other ($\theta$), and then analyse the results in a way that should produce a straight line if the equation is correct. A straight line also lets us obtain constants from a gradient/intercept.

Understanding the Question

You must design a practical method to:

• change $m$ in controlled steps,
• measure the angle $\theta$ at the instant the block just topples,
• keep $V$, $w$, and $L$ constant,
• process the data to check whether the suggested equation fits,
• and explain how to find $\alpha$ from your results.

The apparatus is essentially: a wooden block on a bench, a strip pivoted at the centre of its top, and a hanging mass at a fixed point $P$ on the strip. The strip is rotated in the horizontal plane, and you read $\theta$ in the top view.

Approach

1. Set constants: Measure the block dimensions to compute $V$, and measure $w$. Choose a point on the strip so the load is always a fixed distance $L$ from the nail.
2. Collect data: For a range of masses $m$, find the corresponding “just topples” angle $\theta$. Repeat and average $\theta$ to reduce random uncertainty.
3. Linearise the relationship to the form $y = mx + c$ so that a graph can be plotted. A good choice is to make $\cos\theta$ the dependent variable because the equation already contains $\cos\theta$.
4. Determine $\alpha$ from the gradient (and optionally check consistency using the intercept).

Step-by-Step Reasoning

1) Setting up and measuring fixed quantities

• Measure the block length $l$, width $w$, and height $h$ with a ruler (or vernier calipers for better precision).
• Calculate volume:

$$V = lwh$$

• Mark point $P$ on the strip so that the centre of the hanging load is always at the same distance from the nail. Measure this distance and record it as $L$. Keeping $L$ constant is crucial because it appears in the formula.

A clear apparatus layout also helps you measure $\theta$ consistently.

2) Measuring the toppling angle $\theta$ for each mass $m$

• Attach a mass hanger at $P$ and add slotted masses to set $m$.
• Rotate the strip slowly from a reference direction. The slow rotation reduces overshoot (missing the exact instant of toppling).
• The “just topples” condition is the instant an edge just begins to lift / the block begins to tip.
• Read $\theta$ using a protractor fixed relative to the block/bench so the reference line is repeatable.
• Repeat the measurement for the same $m$ (e.g. 3 repeats) and take the mean $\theta$.
• Do this for at least 6 different masses spanning a wide range (so $1/m$ spans a wide range, improving the gradient determination).

3) Linearising to a straight-line graph

Start from the suggested equation:

$$\alpha Vw = 2mL\cos\theta - mw$$

Bring the $mw$ term to the left:

$$\alpha Vw + mw = 2mL\cos\theta$$

Divide by $2mL$:

$$\cos\theta = \left(\frac{\alpha Vw}{2L}\right)\left(\frac{1}{m}\right) + \frac{w}{2L}$$

This is in the linear form $y = Gx + c$ with:

• $y = \cos\theta$
• $x = 1/m$
• gradient $G = \dfrac{\alpha Vw}{2L}$
• intercept $c = \dfrac{w}{2L}$

So, for each data point you calculate $\cos\theta$ from your measured $\theta$, and calculate $1/m$ from your chosen mass.

4) Extracting $\alpha$

From the gradient definition:

$$G = \frac{\alpha Vw}{2L}$$

Rearrange:

$$\alpha = \frac{2LG}{Vw}$$

So once you have the best-fit straight line and its gradient $G$, you substitute your measured $L$, calculated $V$, measured $w$, and the gradient to obtain $\alpha$.

A useful extra check (not required but good practice) is that the y-intercept from the graph should be close to $w/(2L)$ using your measured $w$ and $L$.

Key Takeaways

• Planning marks come from a clear method: what you vary, what you measure, and what you keep constant.
• Converting a model to a straight-line form is the standard way to test it and to determine constants.
• Here, plotting $\cos\theta$ against $1/m$ should give a straight line if the suggested relationship is correct.
• The constant $\alpha$ is obtained from the gradient using algebra.

Common Mistakes

• Not keeping $L$ constant (moving the load position when changing masses).
• Plotting the wrong variables (e.g. plotting $\theta$ vs $m$ directly, which does not linearise the equation).
• Forgetting to use $\cos\theta$ and trying to fit the original equation without linearisation.
• Not stating how “just topples” is identified (must describe the observation criterion).
• No repeats/averages for $\theta$, leading to large scatter.
• Omitting units for measured lengths and masses.

Things to Be Careful About

• Use $m$ in $\text{kg}$ and lengths in $\text{m}$ for consistency when calculating $\alpha$.
• Ensure the angle reference line is fixed and consistent (mark a line on the block/bench).
• Read the protractor carefully to avoid parallax; rotating slowly reduces the chance of overshooting the toppling point.
• Make sure the hanging mass is free (not touching the bench) and hangs vertically so its position is well-defined.
• Choose masses that cause toppling at measurable angles (avoid angles too close to $0^\circ$ or where toppling becomes sudden and hard to judge).