Physics 9702/52 — May/June 2021

Background Concept

The heater supplies electrical energy to the system at a rate (power) $P$.

If the heater is on for time $t$, the electrical energy supplied is

$$E = Pt.$$

The experiment is based on the idea that this supplied energy is used to raise the temperature of the water and also to warm the metal cylinder (and possibly other parts of the apparatus). In practice there will also be some energy lost to the surroundings, which is why good insulation, a lid, and consistent procedure matter.

The suggested model is

$$Pt = AhW\Delta\theta + Z\Delta\theta,$$

where:

• $A$ is the cross-sectional area of the cylinder,
• $h$ is its height,
• $\Delta\theta$ is the temperature rise of the water,
• $W$ and $Z$ are constants.

This is already close to a straight-line form: it says that for a fixed $\Delta\theta$ (and fixed $h$), $Pt$ depends linearly on $A$.

Understanding the Question

You must design an experiment where you:

1. vary $A$ using different cylinders,
2. measure the corresponding time $t$ for the water temperature to rise by a chosen fixed $\Delta\theta$,
3. keep other variables controlled so that any change in $t$ is mainly due to changing $A$,
4. process the data in a way that tests whether the suggested equation is correct, and
5. use the straight-line graph to find numerical values of the constants $W$ and $Z$.

Because the equation contains $P$, $h$, and $\Delta\theta$, your plan must include how these are measured or kept constant.

Approach

1. Choose $A$ as the independent variable by selecting multiple cylinders of the same material but different diameters.
2. Measure $t$ for a fixed, repeatable temperature rise $\Delta\theta$ (e.g. $10\,^{\circ}\text{C}$). Fixing $\Delta\theta$ makes the timing definition consistent.
3. Measure $P$ (do not assume it is constant unless you verify it): use $P = VI$.
4. Control heat losses and other factors: same mass of water, same container/insulation, same immersion depth, constant stirring.
5. Linearise the suggested equation by dividing through by $\Delta\theta$ to obtain a straight-line relationship of the form $y = mx + c$.
6. Extract constants from gradient and intercept.

Step-by-Step Reasoning

1) Setting up the apparatus

A good arrangement is an insulated beaker (or polystyrene cup) containing water, with the heater and metal cylinder fully submerged. A temperature probe measures the water temperature and a stirrer keeps temperature uniform so the probe reading represents the whole beaker.

Stirring is important: without it, water near the heater becomes hotter than the bulk, so the probe might not record the true average temperature rise, and $t$ becomes inconsistent.

2) Measuring $A$ (and $h$)

The cylinders differ in cross-sectional area, so measure diameter $d$ with vernier calipers (or micrometer) and compute

$$A = \pi \left(\frac{d}{2}\right)^2.$$

Measure at a few positions and average because cylinders may not be perfectly circular.

The equation includes $h$. Ideally use cylinders with the same $h$ (then $h$ is controlled). If heights differ, measure $h$ for each cylinder and either:

• reject cylinders with different $h$, or
• modify the analysis to plot against $Ah$ instead of $A$.

Since the question asks to test a relationship between $t$ and $A$, the cleanest plan is to keep $h$ constant.

3) Measuring power $P$

Power supplied by the heater is

$$P = VI.$$

So include an ammeter in series with the heater and a voltmeter across the heater (or use a power supply with reliable digital readouts). Record $V$ and $I$ for each run; if they vary, use the measured value of $P$ for that run when calculating $Pt/\Delta\theta$.

4) Measuring the time $t$ for a fixed $\Delta\theta$

Choose an initial temperature $\theta_0$ (for example, room temperature) and a fixed temperature rise such as $\Delta\theta = 10\,^{\circ}\text{C}$. Then:

• start the stopwatch when the probe reads $\theta_0$,
• stop it when it reads $\theta_0 + \Delta\theta$.

Repeat each measurement and take an average $t$ to reduce random timing and reading error.

5) Controlling variables

To ensure the model is being tested fairly, keep these constant:

• Water mass: measure the same volume or mass each time.
• Heat loss conditions: use the same insulation and lid; keep the beaker in the same environment.
• Starting temperature: cool back to the same $\theta_0$ between runs (or replace with fresh water at the same $\theta_0$).
• Geometry: keep heater and cylinder positions/depths the same.
• Stirring: same stirring method/rate each run.
• Heater power: same setting and monitor $V$ and $I$.

Good control is essential because unwanted changes (e.g. different heat losses) could create apparent changes in $t$ that are not due to $A$.

6) Linearising and finding $W$ and $Z$

Start from the given relationship:

$$Pt = AhW\Delta\theta + Z\Delta\theta.$$

Divide by $\Delta\theta$:

$$\frac{Pt}{\Delta\theta} = AhW + Z.$$

Now compare with $y = mx + c$:

• Let $y = \dfrac{Pt}{\Delta\theta}$,
• Let $x = A$,
• Then the gradient is $m = hW$ (if $h$ constant),
• The intercept is $c = Z$.

So after plotting the graph:

$$W = \frac{m}{h}, \quad Z = c.$$

A straight line (within scatter) supports the proposed relationship.

Key Takeaways

• A planning question must clearly identify variables, method, and controls.
• Measuring electrical energy uses $E = Pt$ with $P = VI$.
• Linearisation is the key to extracting constants: rearrange into $y = mx + c$ so gradient and intercept give the constants.
• Good experimental design focuses on reducing/controlling heat losses and ensuring uniform water temperature.

Common Mistakes

• Not measuring $P$ (assuming the heater power is exactly constant without checking $V$ and $I$).
• Forgetting to keep $\Delta\theta$ fixed; changing $\Delta\theta$ changes the timing definition and breaks the linear comparison.
• Poor control of starting temperature $\theta_0$, causing inconsistent heat-loss rates and inconsistent $t$.
• No stirring: temperature gradients make the probe reading unreliable.
• Plotting the wrong graph (e.g. plotting $t$ vs $A$ directly without accounting for $P$ and $\Delta\theta$), so you cannot obtain $W$ and $Z$ cleanly.

Things to Be Careful About

• Electrical safety: water and live connections must be kept separate; switch off before adjusting the apparatus.
• Ensure the cylinder is fully submerged each time; partial exposure changes heat transfer.
• Decide how to handle $h$: the simplest is to use cylinders with the same height. If not, you must incorporate $h$ into the analysis (otherwise the gradient no longer equals $hW$).
• When drawing the best-fit line, use a wide range of $A$ values and repeat readings to reduce uncertainty in gradient and intercept.
• Use consistent units (e.g. $A$ in $\text{m}^2$, $P$ in $\text{W}$, $t$ in $\text{s}$, $\Delta\theta$ in $\text{K}$ or $^{\circ}\text{C}$ as a temperature difference).