Physics 9702/52 — February/March 2023

Background Concept

The suggested model links the mechanical/electrical power available from the rotating turbine to the rate at which the pump can lift liquid.

If liquid of mass flow rate $Q$ is lifted through a vertical height $h$, then the gain in gravitational potential energy per unit time (i.e. power) is

$$P_{\text{useful}} = Qgh$$

The relationship provided is

$$Qgh = C + Df^3$$

This has the form of a straight-line equation $Y = C + DX$ if we choose

$$Y = Qgh \qquad \text{and} \qquad X = f^3$$

So, if we can measure $Q$, $h$ and $f$, we can test whether plotting $Qgh$ against $f^3$ gives a straight line, and then use the intercept and gradient to find $C$ and $D$.

Understanding the Question

You are asked to plan (not to calculate) a laboratory experiment that:

• varies the rotation frequency $f$ of a model wind turbine,
• measures the mass flow rate $Q$ of liquid coming out of the top of a vertical pipe,
• uses the measured data to check the relationship $Qgh = C + Df^3$,
• and explains how $C$ and $D$ can be obtained from your results.

You must also include practical details: how to take the measurements, what to keep constant (especially $h$), how to analyse the data, and safety precautions (water + electricity + rotating parts).

Approach

1. Choose variables: make $f$ the independent variable by changing the wind speed from a fan.
2. Measure $Q$: collect the outflow for a known time and measure its mass; then $Q = m/t$.
3. Keep $h$ constant: clamp the outlet at a fixed height above the liquid surface and measure $h$.
4. Linearise: compute $f^3$ and $Qgh$ for each run, then plot $Qgh$ vs $f^3$.
5. Extract constants: from $Y = C + DX$, intercept gives $C$ and gradient gives $D$.
6. Quality: repeat readings to reduce random error and use a wide range of $f$.

Step-by-Step Reasoning

1) Setting up the apparatus

• Place the pump in a container so it stays submerged.
• Connect the pump to the turbine generator.
• Attach the vertical pipe to the pump outlet and clamp it so the outlet position is fixed.

2) Measuring the height $h$

• Define $h$ as the vertical distance from the liquid surface in the container to the outlet at the top of the pipe.
• Measure this with a metre rule (or ruler) held vertically next to the pipe.
• Keep $h$ constant for the whole experiment because the model equation includes $h$; if $h$ changes, the same $f$ would produce a different $Q$ and spoil the test.

3) Producing and varying the frequency $f$

• Use a fan to provide moving air.
• Vary $f$ by adjusting fan speed (preferred) or changing distance/angle.
• Measure $f$ with a tachometer, or by using a light gate/strobe method:
  • tachometer reads directly in $\text{s}^{-1}$ (Hz),
  • or mark one blade and count revolutions in a timed interval, then $f = N/t$ (less precise at high $f$).

4) Measuring the mass flow rate $Q$

• Place a beaker (or measuring jug) at the outlet to collect the liquid.
• Collect for a measured time $t$ using a stopwatch.
• Measure mass $m$ collected using a balance (mass is usually more accurate than reading volume from a measuring cylinder).
• Calculate

$$Q = \frac{m}{t}$$

Units: if $m$ is in kg and $t$ in s, then $Q$ is in $\text{kg s}^{-1}$.

5) Repeats and range

• Take at least 6 different fan settings (so at least 6 different $f$ values), spanning as wide a range as possible.
• Repeat each run (same fan setting) at least twice and average $Q$ to reduce random timing/collection fluctuations.
• Ensure steady flow before starting timing; otherwise the collected mass does not represent the steady-state $Q$.

6) Data processing and graph
For each run compute:

$$X = f^3$$

and

$$Y = Qgh$$

Then plot $Y$ against $X$.

• If the relationship is correct, points lie close to a straight line.
• Draw a best-fit line.
• Use a large triangle to find the gradient:

$$D = \frac{\Delta (Qgh)}{\Delta (f^3)}$$

• Read the $y$-intercept at $f^3 = 0$; this gives $C$.

7) Uncertainties (what to do in a plan)

• Estimate uncertainty in $m$ from balance resolution, in $t$ from reaction time/stopwatch resolution, in $f$ from instrument resolution, and in $h$ from ruler reading.
• Add error bars vertically using uncertainty in $Qgh$ (dominated by uncertainty in $Q$ from $m$ and $t$).
• If required, a worst acceptable line through the error bars can be used to estimate uncertainty in gradient $D$ and intercept $C$.

Key Takeaways

• A planning question is scored for: variables, workable method, clear measurements, control of variables, correct linearisation, and safety.
• Mass flow rate is found from a timed collection: $Q = m/t$.
• To determine two constants, rearrange to a straight-line graph: plot $Qgh$ vs $f^3$ so intercept $= C$ and gradient $= D$.

Common Mistakes

• Plotting $Q$ against $f$ (or even $Q$ against $f^3$) without accounting for $gh$ as required by the given equation.
• Not stating how $Q$ is measured (must specify collecting liquid and timing, then $Q = m/t$).
• Failing to control or even mention that $h$ must be kept constant.
• Using too few readings of $f$ (insufficient to judge linearity) or not repeating measurements.
• Ignoring safety issues with electrical equipment near liquid or exposed rotating blades.

Things to Be Careful About

• Measure $h$ consistently from the liquid surface to the outlet; if the water level changes significantly, either top it up between runs or re-measure $h$ each time.
• Ensure steady-state operation before taking $m$ and $f$ readings.
• Use consistent SI units so that $Qgh$ has units of power (W).
• When finding gradient, use $\Delta y/\Delta x$ with points far apart on the best-fit line, not between adjacent plotted points.
• Keep the fan/turbine position fixed for each run while measurements are taken to avoid $f$ drifting during the timed collection.