Physics 9702/34 — October/November 2013

Background Concept

A height measurement is a length measurement, so you use a ruler/metre rule and quote the value to the smallest reliable division (typically $1\ \text{mm}$ on a standard rule). The dominant issue is often parallax: if your eye is not level with the mark, the reading appears shifted.

Understanding the Question

You are asked to measure $h_1$, the vertical distance from the bench surface to the bottom of the mass hanger when only the hanger is attached to the spring.

Approach

1. Set up the spring and hanger exactly as in the diagram.
2. Place a ruler vertically with its zero at bench level (or measure from bench to hanger directly).
3. Read the position of the bottom of the hanger, with your eye level with the scale.
4. Record $h_1$ with a unit and appropriate precision.

Step-by-Step Reasoning

• Suspend the spring from the clamp rod.
• Attach the mass hanger and allow it to come to rest.
• Hold the ruler next to the hanger so it is vertical.
• Identify the reference point (bottom of the hanger).
• Read the ruler at that point, ensuring your eye is level with the mark.
• Record as, e.g. $h_1 = 32.0\ \text{cm}$ (or in $\text{mm}$), matching the smallest division.

Key Takeaways

• Always state the reference points for a length.
• Avoid parallax and quote a sensible precision.

Common Mistakes

• Measuring to the top of the hanger or to the spring end instead of the bottom of the hanger.
• Not including units.
• Reading the scale with your eye above/below the mark (parallax error).

Things to Be Careful About

• Keep the ruler vertical (otherwise you measure a slanted distance).
• Let the hanger come to rest before reading.
• Record to the nearest $1\ \text{mm}$ (or nearest division of the ruler you actually use).

Background Concept

Adding mass increases the load on the spring so the spring extends and the hanger sits lower, so the height above the bench decreases. The measurement technique is identical to part (a)(i).

Understanding the Question

After adding a $50\ \text{g}$ slotted mass to the hanger, you must measure the new height $h_2$ of the bottom of the hanger above the bench.

Approach

Use the same reference points and the same ruler position as for $h_1$ so that $h_1 - h_2$ is meaningful.

Step-by-Step Reasoning

• Carefully place the $50\ \text{g}$ mass onto the hanger.
• Wait until oscillations have stopped (or gently damp them without changing the equilibrium position).
• Measure from the bench to the bottom of the hanger.
• Record $h_2$ to the same precision as $h_1$ (e.g. nearest $0.1\ \text{cm}$).

Key Takeaways

• Consistency of method (same reference points, same precision) is vital for differences like $h_1-h_2$.

Common Mistakes

• Measuring $h_2$ from a different point (e.g. top of hanger).
• Recording $h_1$ and $h_2$ to different decimal places.

Things to Be Careful About

• Ensure the ruler has not moved relative to the bench reference.
• Read the scale at eye level to reduce parallax.

Background Concept

When you calculate a change in a measured length, you subtract the two readings. The unit stays as a length unit (cm or mm). For exam tables, you should quote the calculated change to a sensible precision consistent with the raw readings.

Understanding the Question

You have measured $h_1$ (before adding the $50\ \text{g}$ mass) and $h_2$ (after adding it). You must calculate:

Approach

Subtract the later height from the initial height. Since adding mass makes the hanger lower, typically $h_2 < h_1$ so $C$ should be positive.

Step-by-Step Reasoning

• Write the definition $C = h_1 - h_2$.
• Substitute your measured values.
• Perform the subtraction.
• Keep the unit (cm, if $h_1$ and $h_2$ were in cm).
• Quote to appropriate precision (often the same decimal place as $h_1$ and $h_2$).

Key Takeaways

• A difference keeps the same unit as the original quantity.
• The sign of the result should make physical sense.

Common Mistakes

• Reversing the subtraction and getting a negative value.
• Dropping the unit for $C$.

Things to Be Careful About

• Use consistent units for $h_1$ and $h_2$ before subtracting.
• Round only at the end of the calculation.

Background Concept

In practical work, correct assembly matters because misalignment (tilted strip, angled springs) changes the forces and therefore changes measured spring extensions. The diagram shows a strip supported at two points by springs; the load position is adjustable.

Understanding the Question

You must assemble the apparatus as shown: two springs support the wooden strip, and a mass $M$ hangs from a string loop that can slide along the strip. Initially the mass should be near the middle so both springs share the load.

Approach

Replicate the geometry in the diagram: springs vertical, attachment points secure, and ensure you can measure $x$ from the left-hand hole to the string loop.

Step-by-Step Reasoning

• Fix two stands on the bench with clamps holding horizontal rods.
• Suspend the left spring from one rod and connect it to the left side of the strip (via the hanger as shown).
• Suspend the right spring from the other rod and connect it to the right side of the strip.
• Place the loop/string around the strip and hang mass $M$ from it.
• Slide the loop so $M$ is close to the centre of the strip.

Key Takeaways

• A correct initial setup reduces systematic errors later.

Common Mistakes

• Attaching the string loop at an angle so the force on the strip is not vertical.
• Springs not in line with the attachment points.

Things to Be Careful About

• Ensure the strip can move freely and is not touching the stands.
• Ensure the string loop can slide smoothly but does not snag.

Background Concept

The relationship you later test assumes the forces are vertical and that the strip is in static equilibrium. If the strip is tilted, the measured spring lengths no longer represent the intended force balance in the same way.

Understanding the Question

Before measuring $x$, $L_A$ and $L_B$, you must make sure the strip is level and the springs are vertical.

Approach

Make small adjustments to the clamp heights/rod positions. Confirm the strip is horizontal by eye (or with a set square/spirit level) and confirm springs are not angled.

Step-by-Step Reasoning

• Slide the stands slightly or adjust clamp heights so the strip looks level.
• If available, place a set square against the strip to check horizontality.
• Ensure each spring hangs straight down (not rubbing against anything).
• Wait for oscillations to stop.

Key Takeaways

• Good alignment reduces systematic error and improves consistency.

Common Mistakes

• Taking measurements while the strip is still oscillating.
• Leaving a spring slightly angled, which changes its effective extension.

Things to Be Careful About

• Do not change the position of the mass along the strip while levelling.
• Ensure both attachment points are at the same horizontal level once adjusted.

Background Concept

A distance along a straight object should be measured with the ruler aligned parallel to that object. If the ruler is at an angle, you introduce a cosine error (measuring a longer diagonal rather than the intended horizontal distance).

Understanding the Question

You need the distance $x$ from the left-hand hole in the strip to the position of the string loop supporting $M$.

Approach

Use the left hole as the zero/reference point, align the ruler with the length of the strip, and read the position of the loop.

Step-by-Step Reasoning

• Place the zero of the ruler at the centre of the left-hand hole (or align a known mark with the hole and subtract).
• Ensure the ruler edge is parallel to the strip.
• Read the position of the string loop (choose a consistent point on the loop, e.g. its centre).
• Record $x$ to the nearest mm (or nearest 0.1 cm).

Key Takeaways

• Define and keep the same reference points for every reading of $x$.

Common Mistakes

• Measuring from the end of the strip rather than the left-hand hole.
• Measuring to the edge of the loop inconsistently as it shifts.

Things to Be Careful About

• Ensure the strip is horizontal before measuring, as specified in (b)(ii).
• Keep the loop stationary while reading.

Background Concept

The spring’s extension is inferred from how the length of its active (coiled) section changes. To compare two springs, you must measure each in the same way (same definition of the ends of the coiled region).

Understanding the Question

You must measure and record:

• $L_A$: coiled length of the left spring
• $L_B$: coiled length of the right spring

Approach

For each spring, identify the top and bottom of the coiled section and measure the distance between these points with the ruler held close and parallel to the spring.

Step-by-Step Reasoning

• Look at the spring and identify where the tight coils start and where they end (exclude hooks if the question indicates only the coiled section).
• Hold the ruler next to the spring.
• Read the positions of the two ends of the coiled section and subtract to get $L$.
• Record $L_A$ and $L_B$ to the nearest mm (or 0.1 cm) consistently.

Key Takeaways

• Measuring a defined section (coils only) improves repeatability.

Common Mistakes

• Including the hook/non-coiled end parts when the instruction says “coiled section”.
• Ruler not aligned with the spring (gives a slightly longer length if diagonal).

Things to Be Careful About

• Keep the spring vertical (as required in (b)(ii)) to avoid measuring a slanted length.
• Read at eye level to reduce parallax.

Background Concept

A good graph is one where the axes are clearly labelled and the scale lets you read values accurately. For Cambridge practical papers, credit is typically given for:

• correct choice of variables on axes,
• sensible, simple scales,
• accurate plotting with small crosses,
• using most of the grid.

Understanding the Question

You must plot a graph with:

• $y$-axis: $\dfrac{(L_A-L_B)}{C}$
• $x$-axis: $x$
  using your six data points.

Approach

Use the table from part (c). Decide your axis ranges from the min/max values. Choose scales such as 1 large square = 1 cm on $x$ or 0.05 on $y$ (whatever fits your data well).

Step-by-Step Reasoning

• Draw axes and label:
  • horizontal: $x/\text{cm}$ (or whatever unit you used),
  • vertical: $\dfrac{(L_A-L_B)}{C}$.
• Choose a scale so your plotted points occupy a large area.
• Plot each of the six points with a small cross ($\times$).
• Check that each point matches the correct $(x, y)$ pair.

Key Takeaways

• Axes labels must include the quantity and unit (unit omitted if dimensionless).
• Scales should be easy to use (1, 2, 5, 10 style steps).

Common Mistakes

• Swapping axes (plotting $x$ on the $y$-axis).
• Missing units on the $x$-axis.
• Using an awkward scale (e.g. 3 units per large square) or using only a small corner of the graph.

Things to Be Careful About

• $\dfrac{(L_A-L_B)}{C}$ is dimensionless, so do not add a unit on that axis.
• Plot points precisely; do not draw large dots.
• Ensure you plotted the ratio column, not $(L_A-L_B)$ itself.

Background Concept

A line of best fit represents the overall trend in data with random scatter. It should not be forced through every point; instead it should reflect the most likely linear relationship.

Understanding the Question

After plotting the six points, you must draw the straight line that best represents them.

Approach

Use a ruler to draw a straight line that passes through the middle of the scatter, aiming for roughly equal numbers of points above and below. Do not simply join neighbouring points.

Step-by-Step Reasoning

• Place the ruler so the line runs through the general centre of the plotted points.
• Adjust so the deviations look balanced.
• Draw one clean straight line across most of the graph.
• If one point is clearly anomalous, the line should still represent the other points.

Key Takeaways

• Best-fit means balanced scatter, not connecting-the-dots.

Common Mistakes

• Joining points with a zig-zag line.
• Forcing the line through the origin when the data do not support it.

Things to Be Careful About

• Draw the line long enough to allow accurate gradient/intercept reading.
• Use a sharp pencil for accuracy.

Background Concept

For a straight-line graph of $y$ against $x$, the equation is:

where:

• $m$ is the gradient $m = \dfrac{\Delta y}{\Delta x}$,
• $c$ is the $y$-intercept (the value of $y$ when $x=0$).

You should find $m$ using two points far apart on the best-fit line to reduce the effect of reading uncertainties.

Understanding the Question

You must determine:

• the gradient of your best-fit line,
• the $y$-intercept of your best-fit line,
  from the graph of $\dfrac{(L_A-L_B)}{C}$ against $x$.

Approach

1. Pick two points on the best-fit line that are widely separated.
2. Read their coordinates carefully.
3. Compute $m = \Delta y/\Delta x$.
4. Find the intercept either by reading where the line crosses the $y$-axis, or by using $c = y - mx$.

Step-by-Step Reasoning

• Draw a large right-angled triangle on the best-fit line.
• Read $x_1, y_1$ and $x_2, y_2$ at the ends of the triangle.
• Calculate:

• Determine the intercept:
  • either read $y$ when $x=0$,
  • or use $c = y_1 - m x_1$.
• State the unit of gradient:
  • $y$ is dimensionless,
  • $x$ is a length,
  • so gradient has unit $\text{(length)}^{-1}$ e.g. $\text{cm}^{-1}$.

Key Takeaways

• Use the best-fit line, not point-to-point gradients.
• Use a large triangle to reduce fractional uncertainty.
• Check units: gradient carries the inverse unit of the $x$-axis.

Common Mistakes

• Using two adjacent plotted points (gives a very uncertain gradient).
• Calculating gradient as $\Delta x/\Delta y$ instead of $\Delta y/\Delta x$.
• Forgetting the unit on the gradient.

Things to Be Careful About

• Do not assume the line passes through the origin.
• Choose points that lie clearly on the best-fit line and are easy to read from grid intersections.
• Keep enough significant figures consistent with graph-reading precision.