Physics 5054/42 — October/November 2012

Walkthrough

Because the ends of the tray slope outwards, the internal length of the tray is shorter at the base than at the top rim. As the depth of water changes, the exact distance the wave travels between the two reflecting walls also changes slightly. Furthermore, a standard straight ruler cannot sit flush against the water surface right up against the sloping wall, making precise alignment challenging.

Key Takeaways

• Irregular or non-vertical geometry introduces uncertainty in defining where a path starts and ends.
• Sloping walls mean distance depends on the height (depth) at which it is measured.

Common Mistakes

• Giving vague answers like "the ruler is too short" or "the water moves".

Things to Be Careful About

• Focus directly on the effect of the sloping ends as stated in the question.

Walkthrough

To overcome the difficulty caused by the sloping ends:

1. Method 1 (Ruler with vertical alignment): Place a rigid ruler above the tray horizontally and sight vertically downwards (at right angles to the ruler) to avoid parallax error when aligning the ends of the water surface with the scale.
2. Method 2 (Set squares): Use set squares placed vertically against the ends at the water level to project the position up to a horizontal ruler.
3. Method 3 (Flexible tape): Use a flexible measuring tape placed directly along the surface of the water between the reflecting ends.

Key Takeaways

• Viewing a ruler perpendicularly prevents parallax error when the ruler cannot touch the object directly.
• Auxiliary tools like set squares or flexible tapes help transfer measurements accurately from awkward geometries.

Common Mistakes

• Not specifying the viewing angle ("look at it") which loses the technique mark.

Things to Be Careful About

• Clearly state how the measurement is taken accurately (e.g., viewing vertically from above).

Walkthrough

Looking at Fig. 1.3, the zero graduation line is not located exactly at the physical edge of the plastic ruler; there is a blank gap (dead space) of several millimetres. If the ruler is stood vertically in the tray with its end resting on the bottom, the water level will be read against the scale, but the measurement will omit the height of this ungraduated gap, leading to an incorrect (underestimated) reading for the depth $d$.

Key Takeaways

• Most standard laboratory rulers have blank plastic margins before the $0\text{ cm}$ mark, which acts as a zero offset when measuring depth directly.

Common Mistakes

• Confusing the dead space with parallax error.
• Simply stating "the ruler starts at zero" without mentioning the physical end of the ruler extends beyond zero.

Things to Be Careful About

• Be precise: refer to the space/gap between the physical end of the ruler and the $0\text{ cm}$ line.

Walkthrough

To obtain the true depth $d$ using two rulers:

1. Measure the offset: Use the graduated scale of Ruler 2 to measure the exact length of the ungraduated end margin (dead space) on Ruler 1.
2. Measure the water level: Stand Ruler 1 vertically on the bottom of the tray and note the reading $h$ where the water surface meets the scale.
3. Calculate total depth: The true depth is:

Alternatively, Ruler 2 can be placed horizontally across the top of the tray and Ruler 1 inverted/lowered, but measuring and adding the end offset is the standard, robust method credited by the mark scheme.

Key Takeaways

• When an instrument has a fixed zero offset, measuring the offset with another instrument and adding/subtracting it yields the correct value.

Common Mistakes

• Forgetting to explicitly state that the measured dead space must be added to the reading on the ruler.

Things to Be Careful About

• Ensure both marking points are met: (1) measure the dead space with the second ruler, (2) add this dead space to the reading.

Walkthrough

To achieve full marks on the graph (4 marks total):

1. Axes (B1): Label the horizontal $x$-axis as $d\text{ / cm}$ and the vertical $y$-axis as $v\text{ / (cm / s)}$.
2. Scales (B1): The question specifies starting at $d = 0$ and $v = 20\text{ cm / s}$.
   • For the $x$-axis, with $d$ going up to $3.0\text{ cm}$, a convenient linear scale is $2\text{ cm}$ (1 large grid square) $= 0.5\text{ cm}$.
   • For the $y$-axis, with $v$ ranging from $20.6$ to $50.5\text{ cm / s}$, a scale of $2\text{ cm}$ (1 large grid square) $= 5\text{ cm / s}$ spans from $20$ to $55\text{ cm / s}$ across 7 large squares, occupying well over half the vertical grid.
3. Plotting points (B1): Each point $(d, v)$ from the table must be plotted with a small neat cross '$\times$' or a dot in a small circle '$\odot$' to within $\frac{1}{2}$ of a small square ($1\text{ mm}$):
   • $(0.5, 20.6)$
   • $(1.0, 28.5)$
   • $(1.5, 35.5)$
   • $(2.0, 42.0)$
   • $(2.5, 46.3)$
   • $(3.0, 50.5)$
4. Line of best fit (B1): Draw a single, continuous, smooth curve that shows a decreasing gradient as $d$ increases. The curve must not be forced into a straight line or drawn with 'point-to-point' jagged segments.

Key Takeaways

• Always check specified starting values for the axes (here: $d=0, v=20$).
• Choose scales that make plotting and reading easy (multiples of 1, 2, or 5).
• Best-fit curves should be drawn in a single smooth motion without kinks, doubling, or feathering.

Common Mistakes

• Forcing a straight line of best fit when the trend is clearly curved.
• Starting the $y$-axis at $0$ instead of the specified $20\text{ cm / s}$, resulting in a compressed scale.
• Plotting large, blurry blobs instead of precise small crosses.

Things to Be Careful About

• Double-check scale intervals: each small square ($2\text{ mm}$) on the $x$-axis represents $0.05\text{ cm}$, and on the $y$-axis represents $0.5\text{ cm / s}$.

Walkthrough

When lifting and lowering the tray to create a wave, water sloshes near the edges. If the depth $d$ exceeds $3.0\text{ cm}$, the water level approaches the rim of the tray, causing water to spill out when the tray is tilted or when the wave reflects at the end walls. Another valid reason is that at larger depths the wave travels too fast, making the travel time too small to measure reliably.

Key Takeaways

• Experimental range is often constrained by physical boundaries of the equipment (dimensions, capacity, overflowing, limits of sensor/human reaction).

Common Mistakes

• Stating that the ruler is not long enough (the ruler is $30\text{ cm}$, which is far greater than $3\text{ cm}$). Focus on the tray or water dynamics.

Things to Be Careful About

• Ensure the suggested limitation is physically realistic based on the apparatus shown in Fig. 1.1.

Walkthrough

Observing the plotted curve on the graph:

• Between $d = 0.5\text{ cm}$ and $d = 1.0\text{ cm}$, the speed increases by $7.9\text{ cm/s}$.
• Between $d = 2.5\text{ cm}$ and $d = 3.0\text{ cm}$, the speed increases by only $4.2\text{ cm/s}$.

The slope (gradient $\frac{\Delta v}{\Delta d}$) is continuously decreasing as depth increases. Extrapolating this trend to very large depths indicates that the rate of increase of speed tends to zero, meaning the speed will level off and approach a constant value (independent of depth in deep water).

Key Takeaways

• A curve flattening out (decreasing slope) indicates that the dependent variable is approaching a plateau or constant value as the independent variable increases.

Common Mistakes

• Claiming the speed will decrease to zero.
• Claiming the speed increases proportionally or infinitely without noticing the flattening curvature.

Things to Be Careful About

• Reference the shape of the graph (gradient decreasing / curve leveling off) to justify the conclusion.