Physics 9702/22 — May/June 2022

Cambridge AS Level · AS Level Structured Questions · worked solutions for every part, with the mark scheme

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Topics Physical Quantities and Units · Work, Energy and Power · Forces, Density and Pressure · Deformation of Solids · Kinematics · Dynamics · +5 more

Q1Physical Quantities and UnitsFree sample

(a)

In the following list, underline all units that are SI base units.

ampere degree Celsius kilogram newton

DifficultyEasy

Worked solution

Answer

SI base units: ampere, kilogram.

Final answer

ampere, kilogram

Detailed explanation

Background Concept

The SI base units are the fundamental units from which all other (derived) units are built. The seven SI base units are:

metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).

A derived unit is formed from base units using multiplication/division and powers, for example

\text{N} = \text{kg m s}^{-2}.

Understanding the Question

You are given a list of units:

ampere, degree Celsius, kilogram, newton
and asked to select (underline) only those that are SI base units.

Approach

Compare each unit in the list with the memorised set of SI base units. If a unit can be written in terms of base units (like N), it is derived and should not be underlined.

Step-by-Step Reasoning

ampere: this is the SI base unit of electric current → underline.
degree Celsius: the SI base unit of temperature is kelvin (K). Degree Celsius is not a base unit (it is related to kelvin by an offset) → do not underline.
kilogram: this is the SI base unit of mass → underline.
newton: unit of force and is derived:

1\ \text{N} = 1\ \text{kg m s}^{-2}

so it is not a base unit → do not underline.

Key Takeaways

Know the seven SI base units.
Units like the newton are derived from base units.
Temperature’s SI base unit is kelvin, not degree Celsius.

Common Mistakes

Underlining newton because it is commonly used (common usage does not mean base unit).
Choosing degree Celsius instead of kelvin as the SI base unit for temperature.

Things to Be Careful About

The question asks specifically for SI base units (not just “SI units”). Some units are SI-derived and still not base.
If unsure, try expressing the unit in base units: if it breaks down, it is not a base unit.

Techniques used

recall the SI base unitsdistinguish base units from derived units and non-SI units

(b)

Fig. 1.1

Fig. 1.1 shows a horizontal beam clamped at one end with a block attached to the other end.

The block is made to oscillate vertically.

The Young modulus $E$ of the material of the beam is given by

E = \frac{kM}{T^2}

where $M$ is the mass of the block,
$T$ is the period of the oscillations
and $k$ is a constant.

A student determines the values and percentage uncertainties of $k$ , $M$ and $T$ .
Table 1.1 lists the percentage uncertainties.

Table 1.1

quantity	percentage uncertainty
$k$	$\pm 2.1\%$
$M$	$\pm 0.6\%$
$T$	$\pm 1.5\%$

The student uses the values of $k$ , $M$ and $T$ to calculate the value of $E$ as $8.245 \times 10^9\ \text{Pa}$ .

(i)

Calculate the percentage uncertainty in the value of $E$ .

percentage uncertainty = ______ $\%$

DifficultyMedium-Easy

Worked solution

Working

Given

E = \frac{kM}{T^2}.

\frac{\Delta E}{E}(\%) = \frac{\Delta k}{k}(\%) + \frac{\Delta M}{M}(\%) + 2\frac{\Delta T}{T}(\%).

= 2.1 + 0.6 + 2(1.5) = 5.7\%.

Answer

$5.7\%$

Final answer

5.7%

Detailed explanation

Background Concept

When quantities are multiplied/divided, their fractional (or percentage) uncertainties add. If a quantity is raised to a power, its fractional uncertainty is multiplied by that power.

For

Q = \frac{ab}{c^n},

the percentage uncertainty is

\frac{\Delta Q}{Q}(\%) = \frac{\Delta a}{a}(\%) + \frac{\Delta b}{b}(\%) + n\frac{\Delta c}{c}(\%).

(The sign from division does not matter because uncertainties are added in magnitude.)

Understanding the Question

You are told

E = \frac{kM}{T^2}

and given percentage uncertainties:

$k$ : $\pm 2.1\%$
$M$ : $\pm 0.6\%$
$T$ : $\pm 1.5\%$

You must find the percentage uncertainty in $E$ .

Approach

Treat $E$ as a product of $k$ and $M$ divided by $T^2$ .

Add the percentage uncertainties of $k$ and $M$ .
Add twice the percentage uncertainty of $T$ (because of the square).

Step-by-Step Reasoning

Start from

E = \frac{kM}{T^2}.

Apply the propagation rule:

\frac{\Delta E}{E}(\%) = \frac{\Delta k}{k}(\%) + \frac{\Delta M}{M}(\%) + 2\frac{\Delta T}{T}(\%).

Substitute:

\frac{\Delta E}{E}(\%) = 2.1 + 0.6 + 2(1.5).

Calculate:

2(1.5) = 3.0,

2.1 + 0.6 + 3.0 = 5.7\%.

Key Takeaways

For multiplication/division: add percentage uncertainties.
For a power $n$ : multiply that term’s percentage uncertainty by $n$ .
Division does not mean “subtract uncertainties”; you still add magnitudes.

Common Mistakes

Subtracting the $T$ uncertainty because $T$ is in the denominator.
Forgetting to multiply the $T$ percentage uncertainty by $2$ .
Using absolute uncertainties without first converting to fractional/percentage uncertainties.

Things to Be Careful About

Make sure the power applies to the uncertainty: $T^2$ means $2(\Delta T/T)$ .
Keep everything consistently in percent since the table is in percent.

Techniques used

use uncertainty propagation rules for products and powersadd percentage uncertainties for multiplied quantitiesmultiply the percentage uncertainty by the power for a powered quantity

(ii)

Use your answer in (b)(i) to determine the value of $E$ , with its absolute uncertainty, to an appropriate number of significant figures.

$E$ = ( ______ $\pm$ ______ ) $\times 10^9\ \text{Pa}$

DifficultyMedium

Worked solution

Working

Percentage uncertainty in $E$ is $5.7\%$ .

Absolute uncertainty:

\Delta E = 0.057 \times 8.245 \times 10^9 = 0.470\times 10^9\ \text{Pa} \approx 0.5\times 10^9\ \text{Pa}.

E = (8.2 \pm 0.5)\times 10^9\ \text{Pa}.

Answer

$(8.2 \pm 0.5)\times 10^9\ \text{Pa}$

Final answer

(8.2 ± 0.5) × 10^9 Pa

Detailed explanation

Background Concept

A percentage uncertainty tells you the size of the uncertainty compared to the value:

\text{percentage uncertainty} = \frac{\Delta x}{x}\times 100\%.

So the absolute uncertainty is

\Delta x = \left(\frac{\text{percentage uncertainty}}{100}\right) x.

When quoting a final result with uncertainty, the usual convention is:

quote the uncertainty to 1 significant figure (sometimes 2 if it begins with 1 or 2),
quote the value to the same decimal place as the uncertainty.

Understanding the Question

From part (i) you have the percentage uncertainty in $E$ .
The calculated value is

E = 8.245\times 10^9\ \text{Pa}.

You must write $E$ in the form

E = (\text{value} \pm \text{absolute uncertainty})\times 10^9\ \text{Pa}

with sensible significant figures.

Approach

Convert $5.7\%$ into a decimal fraction ( $0.057$ ).
Multiply by $8.245\times 10^9\ \text{Pa}$ to get $\Delta E$ .
Round $\Delta E$ appropriately, then round $E$ to match.

Step-by-Step Reasoning

Given percentage uncertainty $= 5.7\%$ :

\frac{\Delta E}{E} = \frac{5.7}{100} = 0.057.

Compute absolute uncertainty:

\Delta E = 0.057 \times 8.245\times 10^9.

First multiply $0.057 \times 8.245$ :

0.057 \times 8.245 = 0.469965 \approx 0.470.

\Delta E \approx 0.470\times 10^9\ \text{Pa}.

Now round the uncertainty to a sensible number of significant figures (typically 1 s.f.):

0.470\times 10^9\ \text{Pa} \approx 0.5\times 10^9\ \text{Pa}.

Because the uncertainty is $0.5\times 10^9$ , the value of $E$ should be rounded to the same decimal place in the bracket (one decimal place):

8.245 \to 8.2.

Therefore,

E = (8.2 \pm 0.5)\times 10^9\ \text{Pa}.

Key Takeaways

Convert percentage uncertainty to absolute uncertainty using $\Delta x = (p/100)x$ .
Quote uncertainty to 1 s.f. (usually), and match the value’s rounding to it.
Keep the power of ten consistent with the requested answer format.

Common Mistakes

Using $5.7$ instead of $0.057$ when finding $\Delta E$ .
Rounding the value but not rounding the uncertainty (or rounding them inconsistently).
Writing $(8.245 \pm 0.5)\times 10^9$ (value has too many digits compared with the uncertainty).
Forgetting the unit $\text{Pa}$ .

Things to Be Careful About

The answer line already includes $\times 10^9\ \text{Pa}$ , so the numbers you write in brackets must be in billions.
Make sure the uncertainty and the value are rounded to the same decimal place within the brackets.

Techniques used

convert percentage uncertainty to absolute uncertaintyapply an uncertainty to a measured/calculated valueround uncertainty and value to appropriate significant figures

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