Physics 9702/53 — October/November 2019

Background Concept

The ball becomes stationary when the upward force from the moving air balances its weight. The suggested model links the ball radius $r$, the equilibrium height $h$, and the motor power $P$:

$$\frac{4\pi r^3 gh}{3} = PK.$$

Here $\frac{4\pi r^3}{3}$ is the volume of a sphere (so the left-hand side contains a factor proportional to volume), $g$ is constant, $P$ is the electrical power delivered to the motor, and $K$ is a constant for the system (expected to be the same for all balls if the model is correct).

To test a proposed relationship experimentally, you typically:

1. identify which variables you can vary and measure reliably,
2. control other factors that could change the outcome,
3. rearrange the relationship into a linear form and use a graph to check for a straight line and determine the constant.

A crucial practical point: motor power is not read directly; for a d.c. motor you can estimate input electrical power using

$$P = VI,$$

where $V$ is the potential difference across the motor and $I$ is the current through it.

Understanding the Question

You must design an experiment where:

• you change the radius $r$ of the ball (by using different balls),
• you measure the height $h$ at which each ball becomes stationary in the air column,
• you measure (or calculate) the motor power $P$,
• you use your measurements to test whether the equation fits the data,
• you explain clearly how to obtain a value for $K$.

The diagram in the question shows $h$ measured vertically from the blower outlet to the centre of the ball, so your measurement method should match that definition.

Approach

A robust way to test the relationship is to make the equation look like the straight-line form $y = mx$:

$$\frac{4\pi r^3 gh}{3} = PK \quad \Rightarrow \quad y = KP,$$

where

$$y = \frac{4\pi r^3 gh}{3}.$$

So if the relationship is correct, a plot of $y$ against $P$ should be a straight line through the origin with gradient $K$.

Practical strategy:

• Use several balls (different $r$) and, for each ball, take several readings at different powers $P$ by adjusting the d.c. supply.
• For each reading, measure $h$, record $V$ and $I$, compute $P$, then compute $y$.
• Combine all data on one graph of $y$ vs $P$: if all points (from different radii) lie on the same straight line through the origin, the model is supported and $K$ is the gradient.

Step-by-Step Reasoning

1. Set up the air column and electrical measurements

• Clamp the blower so it does not move (movement changes the reference for $h$ and can change the flow).
• Put an ammeter in series with the motor circuit and a voltmeter across the motor. This allows $P=VI$ for each setting.

1. Choose and measure the balls (the $r$ measurement)

• Use balls made of the same material and similar finish but different sizes. This is important: if you used different materials, changes in density/drag behaviour could change $h$ even at the same $r$.
• Measure the diameter $d$ with vernier calipers/micrometer. Because balls are not perfectly spherical, measure in two perpendicular directions and average.
• Calculate the radius:

$$r = \frac{d}{2}.$$

1. Collect $h$ data for a range of $P$

• Put the ball into the air stream.
• Adjust the d.c. supply until the ball becomes stationary at a steady height (not oscillating significantly). Wait a short time for it to stabilise.
• Measure $h$ from the top of the blower outlet to the centre of the ball. Reduce parallax by reading the metre rule at eye level and using a set square/pointer aligned to the ball centre.
• Record $V$ and $I$ and compute

$$P = VI.$$

• Repeat for at least 5 different powers for the same ball to give a good spread of points on a graph.
• Repeat the whole set for other radii.

1. Process the data into the required form
   For every row of results, calculate

$$y = \frac{4\pi r^3 gh}{3}.$$

You can now test the relationship by graphing $y$ against $P$.

1. Graph and determine $K$

• Plot $y$ on the vertical axis and $P$ on the horizontal axis.
• Draw a best-fit straight line.
• If the line is (approximately) straight and passes through the origin within scatter, it supports the proportionality $y \propto P$.
• The gradient gives $K$:

$$K = \frac{\Delta y}{\Delta P}.$$

Using many points and a best-fit line is better than calculating $K$ from one pair of readings, because it reduces the impact of random measurement error.

1. Controls and fair test
   Key controlled variables (and why they matter):

• Blower and outlet geometry fixed: changing nozzle size or orientation changes air speed profile, affecting levitation height.
• Ball material/surface the same: drag and stability can depend on surface texture and density.
• Reference level for $h$ fixed: always measure from the same point on the blower outlet.
• Environment: avoid drafts and keep the set-up away from open windows; drafts can alter the ball position.

1. Safety

• Clamp equipment securely to prevent the blower tipping.
• Keep fingers/hair away from moving fan parts and the air intake.
• Use a low-voltage supply and do not exceed the motor’s rated voltage/current; switch off between runs to prevent overheating.

Key Takeaways

• A planning question is about: clear variables, reliable measurements, control of variables, and a graph-based test.
• Convert the given relationship into a linear graph form (here $y = KP$).
• Measure motor power using $P=VI$ and extract the constant from the gradient.

Common Mistakes

• Not measuring $P$ (or treating the power supply setting as “power”): you must use measured $V$ and $I$ because the motor load changes with airflow.
• Only varying $r$ at one power: that tests only $h \propto 1/r^3$ at fixed $P$; it gives a weaker test than using a range of $P$ and a straight-line graph for $K$.
• Measuring $h$ to the top of the ball instead of the centre (inconsistent with the defined $h$).
• Not controlling ball material/surface: different drag behaviour can masquerade as a change in the constant.
• Forgetting repeats: single readings can be affected by oscillations or reading error.

Things to Be Careful About

• Steady height: take readings only when the ball is stationary (or take several readings and average) because fluctuations increase uncertainty in $h$.
• Parallax: $h$ is easy to misread; a pointer/set square improves accuracy.
• Electrical readings: ensure the voltmeter is across the motor (not across the supply leads with significant lead resistance) and the ammeter is in series.
• Units and consistency: use SI units in calculations ($r$ in m, $h$ in m, $P$ in W) so that the computed $K$ has consistent units.
• Graph gradient: use a large triangle and compute $\Delta y/\Delta P$ (not $y/P$ from one point) to minimise gradient uncertainty.