Physics 9702/12 — October/November 2020

Background Concept

Density is defined by

$$\rho = \frac{m}{V}$$

To determine density experimentally you need the mass $m$ and the volume $V$ of the object. However, if a question asks specifically about finding (or estimating) the volume, then only geometric measurements and a suitable geometric model are required.

A coin is commonly approximated as a short cylinder. For a cylinder of radius $r$ and thickness (height) $t$,

$$V = \pi r^2 t$$

Understanding the Question

The student wants the density of the metal, using the coin's volume. The question asks: which item is not needed to estimate the volume of the coin.

The options include diameter, mass, thickness, and the formula for a cylinder's volume.

Approach

1. Decide what shape is used to estimate the coin's volume (a cylinder).
2. Write down the volume formula for that shape.
3. Identify which physical measurements appear in the formula (and therefore are needed).
4. Pick the option that does not appear in, or is not required by, the volume calculation.

Step-by-Step Reasoning

For a cylindrical model,

$$V = \pi r^2 t$$

• To find $r$, you need the diameter (since $r = \tfrac{1}{2}\times \text{diameter}$). So an estimate of diameter is needed.
• You need the thickness $t$ directly. So an estimate of thickness is needed.
• You need to know (or use) the formula for the cylinder volume to combine these measurements. So the cylinder formula is needed.
• The mass $m$ does not appear anywhere in the expression for $V$, so it is not needed to estimate volume.

Therefore the correct option is B.

Key Takeaways

• To estimate an object's volume, you need geometric dimensions and an appropriate geometric formula.
• Mass is needed for density, but not for volume.
• Modelling objects with simple shapes (e.g. coin as a cylinder) is a standard estimation technique.

Common Mistakes

• Choosing mass as needed because the stem mentions density: mass is required for $\rho$ but the question asks about $V$ only.
• Confusing thickness with diameter: both are required for a cylinder.
• Forgetting that you need a formula/model as well as measurements.

Things to Be Careful About

• Read exactly what quantity the question focuses on (here: volume, not density).
• Ensure the chosen model shape matches the object: a coin is well-approximated by a cylinder, so diameter and thickness are the relevant dimensions.