Physics 9702/53 — May/June 2017

Background Concept

The experiment involves an object accelerating from rest, so its speed after moving a known distance depends on the net force along the direction of motion.

The suggested model is

$$\frac{(B + m)v^2}{2s} = Bg - mg\sin\theta$$

where:

• $B$ is the hanging (falling) mass,
• $m$ is the block mass,
• $s$ is the distance moved by the block from $P$ to $Q$ along the plane,
• $v$ is the block speed at $Q$,
• $\theta$ is the angle of the plane to the horizontal,
• $g$ is the gravitational field strength.

A key Paper 5 skill is to test a relationship by making a straight-line graph. To do that, we choose an independent variable we can vary (here $\theta$), measure the dependent variable (here $v$), and rearrange the equation into the form $y = c + Mx$. Then the gradient/intercept allow determination of constants (here $g$).

Understanding the Question

You must design an experiment in which:

• you can set different values of the angle $\theta$,
• the block starts from rest at the same point $P$ each time,
• you can measure the block’s speed $v$ when it reaches a fixed point $Q$ a known distance $s$ from $P$,
• you keep $B$, $m$, and $s$ constant,
• you use your results to find a value for $g$.

The diagram in the stem shows a block on a plane connected by a string over a pulley to a hanging mass. That is the apparatus you should replicate.

Approach

1. Choose variables: vary $\theta$ only; keep $B$, $m$, and $s$ fixed.
2. Measure $v$ at a point: use a light gate at $Q$ with an interrupt card on the block. This gives an instantaneous speed at $Q$ (much better than using average speed over a long interval).
3. Linearise the equation: define

$$y = \frac{(B+m)v^2}{2s}, \quad x = \sin\theta$$

so the model becomes a straight line:

$$y = Bg - mgx.$$

1. Plot and interpret: if the model is correct, the plot of $y$ against $x$ is a straight line. The intercept and gradient each contain $g$, so you can compute $g$.
2. Reduce uncertainty and include safety: repeat runs, average values, and ensure the falling mass is safely caught.

Step-by-Step Reasoning

1. Set up the mechanics:

   • Clamp the pulley firmly at the top of the board.
   • Attach the string from the block over the pulley to the hanging mass.
   • Ensure the string is taut and aligned so the block moves straight along the plane.

2. Fix $P$, $Q$ and $s$ (control variable):

   • Mark $P$ (start) and $Q$ (finish) on the plane.
   • Measure the distance along the plane using a metre rule. Keep this distance constant throughout.

3. Choose a good method to measure $v$ at $Q$:

   • Place a light gate at $Q$.
   • Attach a card of known length $L$ to the block.
   • The data logger measures the blocking time $t$ as the card passes; then

$$v = \frac{L}{t}.$$

This gives the speed at point $Q$ (not just an average between points).

1. Varying $\theta$ (independent variable):

   • Adjust the height of one end of the board to set different angles.
   • Measure $\theta$ directly using a protractor, or indirectly by measuring height $h$ and slope length $\ell$ and using $\sin\theta = h/\ell$.

2. Collecting data:

   • For each angle, release the system from rest with the block at $P$ (no push).
   • Record $t$ at the light gate and calculate $v$.
   • Repeat at least three times for the same $\theta$ and take the mean $v$.
   • Use at least 6 different angles to make a reliable graph.

3. Processing and linear graph:

   • For each angle compute $x = \sin\theta$.
   • Use the measured $v$ to compute

$$y = \frac{(B+m)v^2}{2s}.$$

• Plot $y$ (vertical) against $x$ (horizontal).

1. How the graph gives $g$:
   Since

$$y = Bg - mgx,$$

the straight-line form $y = c + Mx$ has:

• intercept $c = Bg$, so

$$g = \frac{c}{B},$$

• gradient $M = -mg$, so

$$g = -\frac{M}{m}.$$

(You may calculate $g$ both ways and compare; agreement increases confidence.)

1. Uncertainty ideas (what you would do in a real Paper 5 answer):
   • Repeat readings to reduce random error in $t$ (and hence $v$).
   • Use a larger card length $L$ to increase the blocking time, reducing percentage timing uncertainty.
   • Add error bars: uncertainty in $v$ comes from uncertainty in $L$ and $t$; uncertainty in $y$ depends strongly on $v$ because $y \propto v^2$.

Key Takeaways

• A planning question is scored by: sensible apparatus, clear variables, reliable measurements, good control of variables, and a correct graph-based analysis.
• Turning the given equation into a straight-line form is the cleanest way to test the model.
• The gradient and intercept are not just “numbers”: they let you calculate physical constants such as $g$.

Common Mistakes

• Measuring only the time from $P$ to $Q$ and using $v = s/t$ (that gives average speed, not the speed at $Q$ required in the equation).
• Plotting the wrong graph (e.g. $v$ against $\theta$ without linearising, or using $\theta$ instead of $\sin\theta$).
• Not controlling $s$: moving $Q$ between runs changes $y$ and invalidates the test.
• Forgetting to state how $g$ is obtained from the gradient/intercept.
• Too few angles/repeats to justify a straight-line conclusion.

Things to Be Careful About

• Ensure the block is released from rest at the same point $P$ each time; even a small push changes $v$.
• Make sure the hanging mass falls freely and does not hit obstacles; stop it safely.
• Keep $B$ and $m$ constant (do not change the hanging mass when changing $\theta$).
• Use consistent SI units: $m$ and $B$ in $\text{kg}$, $s$ and $L$ in $\text{m}$, giving $y$ in $\text{N}$.
• When calculating $g$ from the gradient, remember the gradient is negative (because $y$ decreases as $\sin\theta$ increases).