Physics 9702/51 — May/June 2023

Background Concept

The relationship given is already in a form that suggests a straight-line graph. In general, if you can write an equation as

$$y = mx + c$$

then plotting $y$ against $x$ should give a straight line whose gradient is $m$ and whose y-intercept is $c$.

Here the suggested relationship is

$$\frac{2d}{t^2} = \frac{AH}{(A + B)} \sin\theta - \frac{KA}{(A + B)}$$

The key idea is:

• you can measure $t$ for different values of $\theta$ while keeping other quantities fixed,
• then you can compute derived quantities ($\sin\theta$ and $2d/t^2$),
• then you use a graph to extract the constants $H$ and $K$ from gradient and intercept.

Understanding the Question

You have a cube on a slope connected by a string over a pulley to a hanging cylinder. You release the cylinder from rest and measure the time $t$ for it to fall a fixed distance $d$.

You are asked to plan an experiment to test how $t$ depends on $\theta$ (the slope angle). Specifically, you must:

• describe what you will vary and what you will measure,
• describe how to keep other factors constant (fair test),
• describe what graph/data processing you will do,
• show how to determine numerical values of two constants, $H$ and $K$.

Approach

1. Choose $\theta$ as the independent variable: set several different angles.
2. For each $\theta$, measure the time $t$ for a fixed fall distance $d$.
3. Convert the equation into the straight-line form by defining
   • $x = \sin\theta$
   • $y = 2d/t^2$
4. Plot $y$ against $x$. If the model is correct, the points lie close to a straight line.
5. Use the measured gradient $m$ and intercept $c$ to calculate $H$ and $K$.
6. Improve reliability: repeat timings, average, use light-gates or data logging, and control frictional conditions.

Step-by-Step Reasoning

1) Decide what to vary and what to measure

• Vary $\theta$ by changing the height of one end of the ramp.
• Measure $t$, the time taken for the cylinder to fall a distance $d$.
• Keep $d$ constant (choose a convenient value and always time over that same distance).

2) Set up a timing method that is accurate and repeatable

The most reliable method is two light-gates with a data logger:

• Put one light-gate at the start position to start the timer when the cylinder begins to fall.
• Put the second light-gate exactly at the position corresponding to a vertical fall of $d$.
• Attach a card/flag to the cylinder so it reliably triggers the gates.

This avoids reaction-time error from a hand-held stopwatch.

3) Control of variables (fair test)

To ensure changes in $t$ are due only to changing $\theta$:

• Use the same cube and cylinder throughout so $A$ and $B$ are constant.
• Use the same string and pulley; ensure the string length is sufficient and remains taut at release.
• Keep the cube on the same face and do not change surfaces (friction affects the motion, and would change the constant $K$ effectively).
• Keep the start position fixed each time (same initial conditions).
• Ensure the pulley rotates freely; check alignment so the string does not rub on the pulley side.

4) Collect sufficient data

For each angle $\theta$:

• record $\theta$ (in degrees or radians, but you will use $\sin\theta$ in calculation),
• time at least 3 repeats and calculate the mean $t$.
  Use at least 6 values of $\theta$ across a sensible range (e.g. not so small that motion is too slow/sticky, and not so large that motion is too fast to measure cleanly).

5) Linearise and plot

Start from

$$\frac{2d}{t^2} = \frac{AH}{(A + B)} \sin\theta - \frac{KA}{(A + B)}$$

Define

$$y = \frac{2d}{t^2}$$

and

$$x = \sin\theta$$

Then

$$y = \left(\frac{AH}{A+B}\right) x + \left(-\frac{KA}{A+B}\right)$$

So the graph of $y$ against $x$ should be a straight line.

• If you do get a straight line within scatter/uncertainty, that supports (tests) the suggested relationship.
• The gradient gives information about $H$.
• The y-intercept gives information about $K$.

6) Extract $H$ and $K$

From the straight-line fit $y = mx + c$:

$$m = \frac{AH}{A+B} \Rightarrow H = m\frac{A+B}{A}$$

and

$$c = -\frac{KA}{A+B} \Rightarrow K = -c\frac{A+B}{A}$$

Measure $m$ and $c$ from the best-fit line (large triangle for $m$), then substitute the measured masses $A$ and $B$.

Key Takeaways

• Good plans identify independent/dependent/control variables clearly.
• Turn the given relationship into $y = mx + c$ so a straight-line graph tests it.
• Choose $x$ and $y$ so that the gradient and intercept map directly to the constants.
• Improve timing reliability (light-gates, repeat readings, average).

Common Mistakes

• Plotting $t$ against $\theta$ directly: the suggested model is linear in $2d/t^2$ versus $\sin\theta$, not in $t$ versus $\theta$.
• Forgetting to keep $d$ constant, which changes $2d/t^2$ independently of $\theta$.
• Not stating how $H$ and $K$ come from gradient/intercept (must show the algebraic link).
• Using too few angles or no repeats, giving weak evidence for a straight-line relationship.
• Stopwatch timing without addressing reaction time (often limits accuracy).

Things to Be Careful About

• Use vertical fall distance $d$ as stated; mark positions carefully and measure $d$ with a ruler/metre rule.
• Ensure release without an initial push (a push changes the initial velocity, affecting $t$).
• Keep frictional conditions unchanged: dust, different cube face, or different ramp surface can change results.
• When calculating $\sin\theta$, ensure the calculator is in the correct angle mode.
• When finding gradient, use a large triangle on the best-fit line and calculate $m = \Delta y/\Delta x$ (not $x/y$).