Physics 9702/52 — May/June 2017

Background Concept

For uniform circular motion, an object moving in a circle of radius $r$ at angular speed $\omega$ needs a centripetal force

$$F_c = m\omega^2 r$$

with $\omega = 2\pi f$. In this experiment, the cube is carried around by the turntable. The cube will only remain at rest relative to the turntable if the available frictional force is large enough to provide the required centripetal force. As the frequency increases, the required centripetal force increases, and at some point the cube starts to slip.

The question supplies a suggested relationship

$$K = 4\pi^2 m f r$$

where $K$ is constant for the cube–surface system being used. Even if you would normally expect $f^2$ in a centripetal-force argument, in Paper 5 you must test the relationship as given: choose measurements and a graph that would confirm proportionalities and allow $K$ to be determined from a gradient.

Understanding the Question

You must design an experiment where you:

• place the cube at a known distance $r$ from the centre,
• find the maximum frequency $f_{\max}$ at which it does not move relative to the turntable,
• repeat for several values of $r$,
• analyse the data to test whether $f$ and $r$ fit the supplied equation, and
• use the graph to obtain a value for $K$.

This is a “threshold” measurement: you are not measuring $f$ at random; you are finding the limiting value just before slipping occurs.

Approach

1. Choose variables: vary $r$ (easy to set precisely) and measure the corresponding $f_{\max}$.
2. Measure $f$ reliably: use an optical sensor/counter with a single reflective marker on the turntable so you get frequency directly in $\text{Hz}$.
3. Define a slip criterion: otherwise different runs give different “maximum” values. A fixed, observable rule (e.g. a certain displacement in a fixed time) makes the threshold reproducible.
4. Linearise the equation: rearrange into $y = mx$ so that a straight-line graph tests the relationship and the gradient gives $K$.
5. Control variables + safety: same cube and surface, consistent placement/orientation, switch off before adjustments, safety screen for flying cube.

Step-by-Step Reasoning

1) Selecting and controlling variables

• Independent variable: $r$.
• Dependent variable: $f_{\max}$.
• Controls:
  • Use the same cube throughout, so $m$ is constant.
  • Use the same turntable surface condition: wipe it clean/dry so frictional properties are steady.
  • Keep the turntable horizontal (spirit level if available). If tilted, a component of weight changes the normal reaction and affects friction.
  • Place the same face of the cube down each time.

2) Measuring $r$

Measure $r$ from the centre of rotation to the centre of the cube (not the nearest edge). Mark radial lines and distances on the turntable; read $r$ to the nearest $1\ \text{mm}$.

3) Measuring $f$ and finding the maximum value before slip

• Put reflective tape on the rim (one piece gives one pulse per revolution).
• Place the photodiode/light gate so it detects each pass; connect to a counter/data logger to read $f$.
• Start at low speed and increase the motor speed slowly.
• Decide a clear slip condition, e.g. “cube moves by at least $1\ \text{mm}$ relative to a mark in $10\ \text{s}$”. This avoids subjective judgement.
• Record $f_{\max}$ as the value just before the slip condition is met.
• Repeat for each $r$ (at least two repeats) and take the mean, because threshold behaviour can vary slightly due to vibration and tiny differences in placement.

4) Linearising and obtaining $K$

Start with the given relationship:

$$K = 4\pi^2 m f r$$

Rearrange to make the dependent variable $f$ the subject:

$$f = \frac{K}{4\pi^2 m}\left(\frac{1}{r}\right)$$

This matches the straight-line form $y = mx$ with:

• $y \equiv f$
• $x \equiv 1/r$
• gradient $G = \frac{K}{4\pi^2 m}$

So:

• plot $f$ against $1/r$;
• a straight line (ideally through the origin) supports the suggested relationship;
• calculate

$$K = 4\pi^2 mG$$

using the measured $m$ and the gradient from the best-fit line.

5) Safety precautions

• Use a safety screen around the turntable or stand well back: the cube can fly off at high speed.
• Switch off the motor before placing/moving the cube.
• Tie back hair and keep loose clothing away from the rotating turntable.
• Increase speed gradually; do not exceed the safe operating speed of the apparatus.

Key Takeaways

• A good plan defines independent/dependent variables, a repeatable measurement criterion, and controls.
• For a suggested relationship, rearrange to linear form and choose a graph where the gradient gives the constant.
• Threshold experiments need careful method: slow changes, clear definition of “just slips”, repeats, and safe working.

Common Mistakes

• Plotting $f$ against $r$ directly (this will not be linear for $f \propto 1/r$).
• Not defining what “does not move” means (makes $f_{\max}$ subjective and inconsistent).
• Measuring $r$ to the edge of the cube rather than to its centre.
• Moving the cube while the turntable is spinning (unsafe and also changes the threshold condition unpredictably).
• Forgetting to measure $m$, so $K$ cannot be found from the gradient.

Things to Be Careful About

• Use SI units consistently when calculating $K$: $m$ in $\text{kg}$, $r$ in $\text{m}$, $f$ in $\text{Hz}$.
• Take a wide range of $r$ values (not clustered) so the graph gives a reliable gradient.
• When taking the gradient, use a large triangle on the best-fit line and compute $\Delta y/\Delta x$.
• Vibrations and airflow can trigger early slipping; keep the apparatus stable and increase speed smoothly.
• Ensure only one reflective marker is used; multiple markers would give an incorrect frequency reading.