Physics 9702/35 — May/June 2014

Cambridge AS Level · Advanced Practical Skills · worked solutions for every part, with the mark scheme

questions

marks

120

minutes

Topics Manipulation, Measurement and Observation · Presentation of Data and Observations · Analysis, Conclusions and Evaluation

Q1Manipulation, Measurement and ObservationPresentation of Data and ObservationsAnalysis, Conclusions and EvaluationFree sample

In this experiment, you will investigate a system in equilibrium due to several forces.

(a)

You have been provided with a wooden beam with 11 holes.

Measure and record the distance $k$ along the wooden beam between the centres of hole 1 and hole 5 as shown in Fig. 1.1.

Fig. 1.1

$k$ = ______

DifficultyEasy

Worked solution

Answer

Measure $k$ between the centres of holes 1 and 5 using a rule.

$k = 12.0\ \text{cm}$

Final answer

12.0 cm

Detailed explanation

Background Concept

In Paper 3, marks for a simple measurement mainly come from good technique and appropriate recording. A length measured with a metre rule should be:

read with the eye normal to the scale (to avoid parallax),
taken between the correct reference points (here, the centres of the holes),
recorded to the precision allowed by the scale (typically to the nearest $1\ \text{mm}$ , i.e. $0.1\ \text{cm}$ ).

Understanding the Question

You are asked to measure the distance $k$ along the beam between the centre of hole 1 and the centre of hole 5 (as indicated by the diagram). The answer space expects a single numerical value with a unit.

Approach

Place the rule along the beam.
Identify the centres of holes 1 and 5.
Read the scale at each centre and subtract (or align one centre with zero and read directly).
Record $k$ with a unit and suitable precision.

Step-by-Step Reasoning

Align the rule so that its scale is parallel to the beam.
Either:
- put the $0$ mark at the centre of hole 1 and read the position of the centre of hole 5, or
- read both positions and calculate the difference.
Record in cm to $0.1\ \text{cm}$ (or in mm), for example $k = 12.0\ \text{cm}$ .

Key Takeaways

Measure between the correct points (centres).
Avoid parallax.
Record with appropriate precision and a unit.

Common Mistakes

Measuring from the edge of a hole instead of its centre.
Not aligning the rule with the beam (introduces systematic error).
Recording without a unit.

Things to Be Careful About

If the $0$ end of the rule is worn, it is better to measure using two readings and subtract.
Make sure the hole centres are judged consistently (use the same reference method each time).

Techniques used

use a rule to measure a length between marked centresavoid parallax by viewing the scale normallyrecord a reading to an appropriate precision

(b)

(i)

Set up the apparatus as shown in Fig. 1.2 with the nail through hole 6 of the wooden beam.

Fig. 1.2

The mass $m$ is $300\text{ g}$ . Position the mass $m$ approximately $15\text{ cm}$ from hole 1.

DifficultyEasy

Worked solution

Answer

Set up the beam and supports as in Fig. 1.2 with the nail through hole 6.

Hang the $300\ \text{g}$ mass so it is approximately $15\ \text{cm}$ from hole 1.

Final answer

Apparatus set up as Fig. 1.2; 300 g hung about 15 cm from hole 1.

Detailed explanation

Background Concept

In equilibrium practicals, the apparatus must be stable and reproducible. The main experimental skill here is correct assembly: ensuring the pivot (nail), beam, spring, and mass are connected exactly as intended.

Understanding the Question

You are instructed to set up the equipment like Fig. 1.2, specifically:

the nail passes through hole 6,
the mass is $300\ \text{g}$ ,
the mass is initially placed about $15\ \text{cm}$ from hole 1.

Approach

Reproduce the diagram: pivot at hole 6, spring-string at hole 1, mass hanging from the beam.
Ensure all clamps are tight and the beam is supported securely.

Step-by-Step Reasoning

Place the beam on the nail through hole 6, held by the clamp stand.
Attach the spring vertically to its support.
Connect the bottom of the spring to the string and attach the string to hole 1 of the beam.
Hang the $300\ \text{g}$ mass from the beam at a point roughly $15\ \text{cm}$ from hole 1 (this is just an initial position; it will be adjusted later).

Key Takeaways

Practical marks come from setting up the correct geometry.
Initial positioning only needs to be approximate if later adjusted.

Common Mistakes

Putting the nail through the wrong hole.
Attaching the spring string to the wrong hole.
Allowing the beam to rub against the clamp (adds friction and affects equilibrium).

Things to Be Careful About

Make sure the mass hangs freely and does not touch the bench.
Check that strings are taut and not snagged on the beam.

Techniques used

assemble apparatus as shown in a diagramposition a load at a specified approximate distancesecure supports to maintain a stable set-up

(ii)

Adjust the apparatus so that the spring is vertical and the wooden beam is horizontal.

The distance $a$ is the distance between the nail and the string attached to the spring.

The distance $b$ is the distance between the nail and the string attached to the mass as shown in Fig. 1.3.

Fig. 1.3

DifficultyMedium-Easy

Worked solution

Answer

Adjust until:

the spring is vertical
the wooden beam is horizontal.

Final answer

Spring vertical; beam horizontal.

Detailed explanation

Background Concept

For equilibrium measurements, the geometry matters. If the spring is not vertical or the beam is not horizontal:

distances $a$ and $b$ are no longer the intended perpendicular lever arms,
extra components of force and unwanted torques can appear,
repeatability is reduced.

Understanding the Question

You must adjust the apparatus so that the spring is vertical and the beam is horizontal before measuring distances.

Approach

Move the positions of clamps/stands and (later) the hanging mass to achieve the required orientation.
Re-check alignment after each adjustment.

Step-by-Step Reasoning

View the spring against a vertical reference (e.g. a retort stand) and adjust until it hangs straight down.
View the beam and adjust the mass position or supports until the beam is level.
Ensure nothing is moving and the beam is at rest before taking readings.

Key Takeaways

Correct alignment reduces systematic errors.
Always align first, then measure.

Common Mistakes

Measuring $a$ and $b$ while the beam is slightly tilted.
Assuming the spring is vertical without checking (it can be pulled sideways by the string).

Things to Be Careful About

After moving the mass or nail, the system may rotate slightly; re-check vertical/horizontal each time.
Make sure the spring is not twisting or caught, which can pull it away from vertical.

Techniques used

adjust apparatus to satisfy alignment conditionscheck horizontality and verticality by eye or using a set squarereduce systematic errors by ensuring correct geometry

(iii)

Measure and record $a$ and $b$ .

$a$ = ______
$b$ = ______

DifficultyMedium-Easy

Worked solution

Answer

Measure distances on the horizontal beam.

$a = 15.8\ \text{cm}$

$b = 26.3\ \text{cm}$

Final answer

a = 15.8 cm, b = 26.3 cm

Detailed explanation

Background Concept

Distances such as $a$ and $b$ are measured along the beam and are used later to form derived quantities ( $1/b$ and $a/b$ ). Small errors in $a$ or $b$ can noticeably affect a graph because $1/b$ is sensitive to changes in $b$ .

Understanding the Question

You must measure:

$a$ : the distance from the nail (pivot) to the string attached to the spring,
$b$ : the distance from the nail (pivot) to the string attached to the mass,
as shown in Fig. 1.3.

Approach

Ensure the beam is horizontal and at rest.
Use a rule to measure along the beam between the correct points.
Record both values with consistent precision.

Step-by-Step Reasoning

Identify the pivot position (nail through the beam).
Identify the vertical line of the spring-string attachment point and the vertical line of the mass-string attachment point.
Measure the horizontal distances from the pivot to each point.
Record, for example: $a = 15.8\ \text{cm}$ and $b = 26.3\ \text{cm}$ (to $0.1\ \text{cm}$ if using a mm scale).

Key Takeaways

Measure between the correct reference points.
Use consistent units and precision.

Common Mistakes

Measuring from the wrong side of the nail (using an incorrect reference point).
Measuring diagonally instead of along the beam.
Recording $a$ and $b$ to different precisions.

Things to Be Careful About

If the string has thickness, measure to its centre line for consistency.
Re-check that the beam has not moved while measuring.

Techniques used

measure horizontal distances between reference linesrecord readings to an appropriate precision with unitsrepeat or check readings for consistency

(iv)

Measure and record the length $L$ of the stretched spring as shown in Fig. 1.4.

Fig. 1.4

$L$ = ______

DifficultyEasy

Worked solution

Answer

Measure the length of the coiled part of the stretched spring.

$L = 8.5\ \text{cm}$

Final answer

8.5 cm

Detailed explanation

Background Concept

When a spring is stretched, its length (or extension) is linked to the force it provides. In this experiment, later readings are taken while keeping $L$ the same, so that the spring force is kept (approximately) constant.

Understanding the Question

You must measure $L$ , the length of the coiled part of the stretched spring (as indicated in Fig. 1.4), and record it.

Approach

Read the spring length between the two indicated endpoints.
Record to a sensible precision.

Step-by-Step Reasoning

Identify the top and bottom of the coiled section (not including hooks if the diagram indicates coil-only).
Place the rule close to the spring and read at eye level.
Record, for example, $L = 8.5\ \text{cm}$ .

Key Takeaways

Measure exactly what the diagram defines as $L$ .
Keep the same definition of endpoints throughout.

Common Mistakes

Measuring including the hook when $L$ is defined for the coiled section only.
Allowing the rule to be at an angle to the spring.

Things to Be Careful About

Make sure the spring is vertical; otherwise the measured length can be misleading.
Avoid parallax when reading the scale.

Techniques used

measure the extension length of a spring between defined endpointsread a scale without parallaxrecord a measurement with a unit and suitable precision

(c)

Vary $a$ by moving the nail to a different hole.

Adjust $b$ until the value of $L$ is the same as in (b)(iv).

Ensure that the spring is vertical and the beam is horizontal.

Measure and record $a$ and $b$ .

$a$ = ______
$b$ = ______

DifficultyMedium-Easy

Worked solution

Answer

Move the nail to a different hole to change $a$ .
Adjust the position of the mass until $L$ is the same as in (b)(iv).
Ensure spring vertical and beam horizontal.

Record, e.g.

$a = 9.5\ \text{cm}$

$b = 23.8\ \text{cm}$

Final answer

Example: a = 9.5 cm, b = 23.8 cm (with L kept the same)

Detailed explanation

Background Concept

To investigate a relationship between two distances, you must vary one variable systematically while controlling others. Here, you change $a$ (by moving the nail) and then adjust $b$ so that $L$ stays constant. Keeping $L$ constant helps keep the spring force the same from run to run.

Understanding the Question

You must:

change $a$ by choosing a different hole for the nail,
then adjust $b$ until the spring length $L$ matches the earlier value,
then measure and record the new $a$ and $b$ .

Approach

Move nail to a new hole (new pivot position) → $a$ changes.
Slide the mass along the beam until the spring length returns to the original $L$ .
Re-check spring vertical and beam horizontal.
Measure $a$ and $b$ and record with units.

Step-by-Step Reasoning

After moving the nail, the beam will not be in equilibrium immediately; adjust the mass position.
Compare the spring length to the previous $L$ reading; adjust until it matches.
Once stable, measure $a$ and $b$ along the beam.
Record a pair of readings (example values shown in the solution).

Key Takeaways

Vary $a$ systematically.
Keep $L$ constant as a control condition.
Measure only once the system is stationary and aligned.

Common Mistakes

Not returning $L$ to the original value before measuring $a$ and $b$ .
Forgetting to re-level the beam and re-verticalise the spring after adjustments.

Things to Be Careful About

Judge $L$ consistently (same endpoints and same viewing angle each time).
Small changes in $b$ may significantly change equilibrium; adjust gently and allow oscillations to stop.

Techniques used

change the independent variable by repositioning apparatusadjust a dependent position to keep a control variable constantmeasure and record a new pair of readings

(d)

Repeat (c) until you have six sets of readings of $a$ and $b$ .

Include values of $\frac{1}{b}$ and $\frac{a}{b}$ in your table.

10M

DifficultyMedium

Worked solution

Answer

Take six sets of readings of $a$ and $b$ (with $L$ unchanged), and calculate $1/b$ and $a/b$ .

Example of a suitable table:

$a$ / $\text{cm}$	$b$ / $\text{cm}$	$1/b$ / $\text{cm}^{-1}$	$a/b$
4.4	21.7	0.0461	0.202
9.5	23.8	0.0420	0.399
15.8	26.3	0.0380	0.601
23.5	29.4	0.0340	0.799
33.3	33.3	0.0300	1.00
46.2	38.5	0.0260	1.20

Final answer

Table of 6 readings of a and b with calculated 1/b and a/b (see working).

Detailed explanation

Background Concept

A good practical table must allow someone else to understand and use your data. The required derived quantities here are:

\frac{1}{b}

and

\frac{a}{b}

If $a$ and $b$ are measured in cm, then $1/b$ has unit $\text{cm}^{-1}$ , while $a/b$ is dimensionless.

Understanding the Question

You must repeat part (c) until you have six sets of readings. You then need one clear table containing:

measured $a$ and $b$ ,
calculated $1/b$ ,
calculated $a/b$ .

Approach

Choose at least six different nail positions to vary $a$ over a reasonable range.
For each, adjust the mass position until the spring length $L$ matches the original value.
Measure $a$ and $b$ , then compute $1/b$ and $a/b$ .
Present all values in one table with correct headings and units.

Step-by-Step Reasoning

Collect six readings:
- Each run: new $a$ (new nail hole) → adjust $b$ until $L$ matches → measure $a$ and $b$ .
Calculate columns for each row:

\frac{1}{b} \text{ (with unit)}

and

\frac{a}{b} \text{ (no unit)}

Presentation requirements:
- Put the unit in the heading, e.g. $b/\text{cm}$ .
- Keep consistent precision for $a$ and $b$ (e.g. to $0.1\ \text{cm}$ ).
- Quote $1/b$ to a sensible number of significant figures (often 3 s.f.).

Key Takeaways

A complete table has quantity, symbol, and unit in headings.
Derived quantities must be calculated correctly with correct units.
Use a good range of $a$ values to improve graph reliability.

Common Mistakes

Missing units in the headings (e.g. writing just " $a$ " instead of " $a/\text{cm}$ ").
Writing $1/b$ without unit.
Inconsistent decimal places within a column.
Calculating $a/b$ incorrectly (e.g. $b/a$ ).

Things to Be Careful About

Since $1/b$ is a reciprocal, small errors in $b$ can affect $1/b$ noticeably; measure $b$ carefully.
Ensure $L$ is genuinely the same in every run; otherwise the relationship you graph may not be linear.

Techniques used

collect multiple sets of readings across a range of the independent variableconstruct a results table with quantity and unit headingscalculate derived quantities from measured datause consistent significant figures and decimal places within columns

(e)

(i)

Plot a graph of $\frac{1}{b}$ on the $y$ -axis against $\frac{a}{b}$ on the $x$ -axis.

DifficultyMedium

Worked solution

Answer

Plot $y = 1/b$ ( $\text{cm}^{-1}$ ) against $x = a/b$ .

Label axes: $x$ -axis $a/b$ (no unit), $y$ -axis $1/b\ /\ \text{cm}^{-1}$ .
Use a suitable scale (at least half the grid in both directions).
Plot all six points accurately.

Final answer

Graph of 1/b (y) against a/b (x) plotted with correct labels and scales.

Detailed explanation

Background Concept

Graph marks in Paper 3 depend on:

correct choice of variables on each axis,
correct axis labels with units,
sensible scales (not cramped; not awkward like 3 squares = 1 unit),
accurate plotting (small, neat crosses/dots).

Understanding the Question

You must plot a graph with:

$y$ -axis: $1/b$ ,
$x$ -axis: $a/b$ .
You will use your table from (d).

Approach

Compute each $(a/b,\ 1/b)$ pair from the table.
Choose axis ranges that include all points.
Label with correct units.
Plot each point carefully.

Step-by-Step Reasoning

diagram-e-graph

$a/b$ is dimensionless, so label the $x$ -axis as $a/b$ .
If $b$ is in cm then $1/b$ is in $\text{cm}^{-1}$ , so label the $y$ -axis as $1/b\ /\ \text{cm}^{-1}$ .
Use a scale that spreads the data out (typically using at least half the graph paper in each direction).
Plot all six points from your table.

Key Takeaways

Put the unit in the axis label for $1/b$ .
Use good scales and neat plotting to gain full marks.

Common Mistakes

Swapping axes (plotting $a/b$ on the $y$ -axis).
Missing the unit on $1/b$ .
Using a poor scale so points occupy only a small corner.

Things to Be Careful About

Plotting accuracy: use a sharp pencil and small crosses.
Make sure you plot each row from your table correctly; one transposed value can spoil linearity.

Techniques used

choose sensible axis scales to use most of the gridlabel axes with quantities and unitsplot experimental points accurately from a table

(ii)

Draw the straight line of best fit.

DifficultyMedium-Easy

Worked solution

Answer

Draw a single straight line of best fit through the plotted points (approximately equal scatter of points on either side).

Final answer

Straight line of best fit drawn.

Detailed explanation

Background Concept

A best-fit line is not drawn point-to-point. For experimental scatter, the best-fit line should represent the overall trend so that points are roughly balanced above and below the line.

Understanding the Question

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

Approach

Use a ruler.
Place the line so the total scatter is balanced.
Extend the line over the full range of plotted points.

Step-by-Step Reasoning

Visually judge the trend (here it should be a straight line with negative gradient).
Position the ruler and draw one thin straight line.
Do not force the line through every point; allow for random scatter.

Key Takeaways

One straight, thin best-fit line.
Balanced scatter.

Common Mistakes

Joining dots.
Forcing the line through an outlier when most points follow a different trend.

Things to Be Careful About

The line should be long enough to allow accurate gradient/intercept reading later.
Keep the line thin; thick lines reduce reading accuracy.

Techniques used

draw a single straight line that balances scatter about the trendignore anomalous points only if clearly inconsistentextend the best-fit line across the full data range

(iii)

Determine the gradient and $y$ -intercept of this line.

gradient = ______
$y$ -intercept = ______

DifficultyMedium

Worked solution

Working

Using two well-separated points on the best-fit line, e.g.
$(x_1, y_1) = (0.20,\ 0.046\ \text{cm}^{-1})$ and $(x_2, y_2) = (1.20,\ 0.026\ \text{cm}^{-1})$ .

\text{gradient} = \frac{\Delta y}{\Delta x} = \frac{0.026 - 0.046}{1.20 - 0.20} = -0.020\ \text{cm}^{-1}

$y$ -intercept from line (at $x=0$ ):

\text{intercept} = 0.050\ \text{cm}^{-1}

Answer

gradient $= -0.020\ \text{cm}^{-1}$

$y$ -intercept $= 0.050\ \text{cm}^{-1}$

Final answer

gradient = −0.020 cm⁻¹, y-intercept = 0.050 cm⁻¹

Detailed explanation

Background Concept

For a straight-line graph of the form:

y = mx + c

the gradient is:

m = \frac{\Delta y}{\Delta x}

and the y-intercept is $c$ , the value of $y$ when $x = 0$ .

Units:

Here, $x = a/b$ is a ratio of lengths, so it has no unit.
$y = 1/b$ has unit of reciprocal length, e.g. $\text{cm}^{-1}$ (or $\text{m}^{-1}$ if using metres).
So the gradient has the same unit as $y$ .

Understanding the Question

You must find two numerical quantities from your drawn best-fit line:

the gradient,
the y-intercept.
These will be used later to calculate constants.

Approach

Use a large triangle on the best-fit line (choose two points far apart to reduce percentage reading error).
Compute $\Delta y$ and $\Delta x$ and divide.
Read (or calculate) the y-intercept at $x=0$ .

Step-by-Step Reasoning

Pick two points on the line (not necessarily measured points) that lie exactly on grid intersections if possible.
Read their coordinates carefully.
Calculate:

m = \frac{y_2 - y_1}{x_2 - x_1}

Because the line slopes downwards, the gradient should be negative.
For the y-intercept, extend the line to meet the y-axis and read the value at $x=0$ .

Key Takeaways

Gradient is always $\Delta y/\Delta x$ , using two well-separated points.
Intercept is read where the best-fit line crosses the y-axis.
Include correct units for gradient and intercept.

Common Mistakes

Using $\Delta x/\Delta y$ (inverting the gradient).
Using two points that are too close together (large percentage uncertainty).
Finding the intercept from a data point rather than from the best-fit line.
Forgetting the negative sign on a downward-sloping graph.

Things to Be Careful About

Use consistent units: if $b$ is in cm, keep $1/b$ in $\text{cm}^{-1}$ throughout.
Read values to a sensible precision based on your graph scale (do not overstate precision).

Techniques used

use a large triangle on the best-fit line to find gradientcalculate gradient as $\Delta y / \Delta x$read the y-intercept from the fitted line

(f)

The quantities $a$ and $b$ are related by the equation

\frac{1}{b} = -\frac{Pa}{b} + Q

where $P$ and $Q$ are constants.

Use your answers in (e)(iii) to determine the values of $P$ and $Q$ .
Give appropriate units.

$P$ = ______
$Q$ = ______

DifficultyMedium-Easy

Worked solution

Working

Given

\frac{1}{b} = -\frac{Pa}{b} + Q

Let $y = 1/b$ and $x = a/b$ :

y = -P x + Q

So gradient $= -P$ and y-intercept $= Q$ .

P = -\text{gradient} = -(-0.020\ \text{cm}^{-1}) = 0.020\ \text{cm}^{-1}

Q = 0.050\ \text{cm}^{-1}

Answer

$P = 0.020\ \text{cm}^{-1}$

$Q = 0.050\ \text{cm}^{-1}$

Final answer

P = 0.020 cm⁻¹, Q = 0.050 cm⁻¹

Detailed explanation

Background Concept

Most Paper 3 graph analysis is about recognising the straight-line form:

y = mx + c

and matching your plotted variables to $x$ and $y$ so you can identify:

gradient $m$ ,
intercept $c$ ,
then relate these to physical constants.

Understanding the Question

You are given:

\frac{1}{b} = -\frac{Pa}{b} + Q

You have already plotted $1/b$ against $a/b$ , and you have found the gradient and y-intercept. You must now use those graph values to find $P$ and $Q$ and give units.

Approach

Rewrite the given equation so it looks exactly like $y = mx + c$ .
Identify which symbol corresponds to the gradient and which corresponds to the intercept.
Use the sign carefully: the gradient equals $-P$ .
Work out units from the fact that $a/b$ is dimensionless and $1/b$ has unit of reciprocal length.

Step-by-Step Reasoning

Define the plotted variables:

y = \frac{1}{b}, \quad x = \frac{a}{b}

Substitute into the given equation:

y = -P x + Q

Compare with $y = mx + c$ :

m = -P, \quad c = Q

Hence:

P = -m

and

Q = c

Units:

$x=a/b$ has no unit.
$y=1/b$ has unit $\text{cm}^{-1}$ (or $\text{m}^{-1}$ ).
Therefore both $m$ and $Q$ have unit $\text{cm}^{-1}$ , and so does $P$ .

Key Takeaways

Always rewrite into $y=mx+c$ using the variables you actually graphed.
The negative sign matters: here $P$ is the negative of the gradient.
Units come from the axis units.

Common Mistakes

Taking $P$ equal to the gradient instead of the negative of the gradient.
Giving $P$ no unit (it does have units here).
Mixing cm and m without converting (e.g. using $Q$ in $\text{cm}^{-1}$ with $k$ in m).

Things to Be Careful About

If your gradient is negative (as expected), $P$ should come out positive.
Use a consistent unit system for later calculations (especially part (g)).

Techniques used

compare an experimental straight-line graph with y = mx + cidentify constants from gradient and interceptdeduce units of constants from the plotted quantities

(g)

The mass $M$ of the wooden beam is given by

M = \frac{m}{kQ}

Use values in (a), (b)(i) and (f) to determine the value of $M$ .
Include a unit for $M$ .

$M$ = ______

DifficultyMedium-Easy

Worked solution

Working

M = \frac{m}{kQ}

$m = 300\ \text{g} = 0.300\ \text{kg}$

$k = 12.0\ \text{cm} = 0.120\ \text{m}$

$Q = 0.050\ \text{cm}^{-1} = 5.0\ \text{m}^{-1}$

M = \frac{0.300}{(0.120)(5.0)} = 0.500\ \text{kg}

Answer

$M = 0.500\ \text{kg}$

Final answer

0.500 kg

Detailed explanation

Background Concept

When a formula involves experimental constants (here $Q$ from a graph and $k$ from a measurement), full marks depend on:

correct substitution,
consistent units,
a final answer with the correct unit.

Understanding the Question

You are given:

M = \frac{m}{kQ}

You must use:

$m = 300\ \text{g}$ (given in the instructions),
your measured $k$ from (a),
your value of $Q$ from (f),
to calculate $M$ .

Approach

Convert all quantities to a consistent unit system (preferably SI: kg, m, $\text{m}^{-1}$ ).
Substitute into the formula.
Check units: $kQ$ is dimensionless because $k$ has unit m and $Q$ has unit $\text{m}^{-1}$ .

Step-by-Step Reasoning

Write the formula clearly:

M = \frac{m}{kQ}

Convert units:

$m$ : $300\ \text{g} = 0.300\ \text{kg}$ .
$k$ : if measured in cm, convert to m by dividing by 100.
$Q$ : if in $\text{cm}^{-1}$ , convert to $\text{m}^{-1}$ by multiplying by 100 (because $1\ \text{cm}^{-1} = 100\ \text{m}^{-1}$ ).

Substitute and calculate.
Quote $M$ with unit kg (or g if you stayed consistently in g and cm).

Key Takeaways

Convert $Q$ carefully: reciprocal units can be tricky.
Check that your final unit is a mass unit.

Common Mistakes

Using $k$ in cm while using $Q$ in $\text{m}^{-1}$ (or vice versa).
Converting $\text{cm}^{-1}$ to $\text{m}^{-1}$ the wrong way round.
Omitting the unit for $M$ .

Things to Be Careful About

If you keep everything in cm and g, that is fine provided you keep it consistent:
- $m$ in g, $k$ in cm, $Q$ in $\text{cm}^{-1}$ gives $M$ in g.
Do not over-round intermediate values; round at the end to an appropriate number of significant figures.

Techniques used

substitute experimental values into a given formulaconvert units consistently before calculationreport a final value with an appropriate unit

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Q2Manipulation, Measurement and Observation · Analysis, Conclusions and Evaluation · Presentation of Data and Observations20M

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