Derived Units & Homogeneity | CAIE 9702 Paper 1 Notes

Paper 1

Physics9702

Derived Units & Homogeneity

Some quantities are measured directly. Others are calculated from measured quantities. Their units must show the same calculation.

Replacing every symbol with base units lets you build a derived unit and check whether an equation can be correct.

Building a derived unit

A is any unit built from base units.
To find one, start from the quantity's defining equation and replace each symbol with its base units.

A dynamics trolley travels along a straight track. A metre rule and a stopwatch are ready to measure its distance and travel time. — Distance and time are measured directly. Speed is calculated from them, so m s⁻¹ is a derived unit.

The trolley's distance is measured in metres and its time is measured in seconds. Therefore the calculated speed has the unit:

\text{unit of speed} = \frac{\text{m}}{\text{s}} = \text{m s}^{-1}

Quantity	Start from	SI base units
force	F = ma	kg m s⁻²
energy	E = Fd	kg m² s⁻²
pressure	p = F / A	kg m⁻¹ s⁻²

These examples use relations taught later in the course. At this point, a question may supply the relation; focus on replacing each quantity with its base units.

Learn the usual named units, but do not guess their base-unit form. Start from a defining equation, substitute base units and simplify.

Worked example

Base units of power

Definition: $\text{power} = \dfrac{\text{work}}{\text{time}}$ .
Work has the units of energy: $\text{kg m}^2\,\text{s}^{-2}$ .
Divide by time: $\dfrac{\text{kg m}^2\,\text{s}^{-2}}{\text{s}} = \text{kg m}^2\,\text{s}^{-3}$ .

Answer

\text{power} = \text{kg m}^2\,\text{s}^{-3}

(9702/22/F/M/24 Q1(b))

Your turn9702/13/O/N/21 Q2 · 1 mark · adapted

Express the ohm in SI base units. Use resistance = potential difference ÷ current and potential difference = power ÷ current.

Show worked answer

First the volt: $[V] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}$ .
Then the ohm: $[R] = \dfrac{[V]}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}$ .

Answer

\text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}

Checking an equation: homogeneity

A physical equation is when every term has the same base units. A term is one complete part separated by $+$ , $-$ or the equals sign.
If even one term has different units, the equation is wrong. Units cannot be added to or equated with a different kind of quantity.
Matching units cannot prove an equation is right. A pure number like $\tfrac12$ or $2\pi$ has no units, so this check cannot see it.
In $s = vt$ , $s$ is displacement, $v$ is velocity and $t$ is time. The left side has unit m and the right side reduces to the same unit:

\text{m s}^{-1} \times \text{s} = \text{m}

The units match, so the equation passes the homogeneity check. This does not prove that the numerical factor or physical model is correct.

Worked example

Testing an equation for homogeneity

A mass hangs on a spring with spring constant $k$ . Test whether $T = 2\pi\sqrt{m/k}$ is homogeneous.

Find the base units of $k$ from $F = kx$ : $[k] = \dfrac{\text{kg m s}^{-2}}{\text{m}} = \text{kg s}^{-2}$ .
Then $\left[\dfrac{m}{k}\right] = \dfrac{\text{kg}}{\text{kg s}^{-2}} = \text{s}^2$ .
The square root gives $\sqrt{\text{s}^2} = \text{s}$ , which matches $T$ . The $2\pi$ has no units, so the equation is homogeneous.

Answer

\text{homogeneous}

(9702/12/F/M/24 Q3)

Worked example

One change: find the unknown power

$T = 2\pi(L/g)^{n}$ is homogeneous. Find $n$ .

$\left[\dfrac{L}{g}\right] = \dfrac{\text{m}}{\text{m s}^{-2}} = \text{s}^2$ .
Right side: $(\text{s}^2)^{n} = \text{s}^{2n}$ . Left side: $\text{s}$ .
Match the powers: $2n = 1$ , so $n = \tfrac{1}{2}$ .

Answer

n = \tfrac{1}{2}

(9702/12/M/J/25 Q4)

Worked example

One more change: base units of a constant

The power output of a star is $P = \sigma A T^4$ , where $A$ is area and $T$ is temperature. Find the base units of the constant $\sigma$ .

Make the constant the subject: $\sigma = \dfrac{P}{A T^4}$ .
Substitute base units:
$[P]=\text{kg m}^2\,\text{s}^{-3}$
$[AT^4]=\text{m}^2\,\text{K}^4$
$[\sigma]=\text{kg s}^{-3}\,\text{K}^{-4}$

Answer

[\sigma] = \text{kg s}^{-3}\,\text{K}^{-4}

(9702/13/O/N/24 Q3)

These three examples are the three ways the exam asks it: test an equation, find an unknown power, or find the base units of a constant. All three use the same skill: they replace each symbol with its base units, then compare.

Common mistake

Do not give units to a pure-number constant (

2\pi

and

4\pi^2

have none). And when one side has several terms, reduce every term, not just the first one.

Derived Units & Homogeneity

Some quantities are measured directly. Others are calculated from measured quantities. Their units must show the same calculation.

Replacing every symbol with base units lets you build a derived unit and check whether an equation can be correct.

Building a derived unit

A is any unit built from base units.
To find one, start from the quantity's defining equation and replace each symbol with its base units.

The trolley's distance is measured in metres and its time is measured in seconds. Therefore the calculated speed has the unit:

\text{unit of speed} = \frac{\text{m}}{\text{s}} = \text{m s}^{-1}

Quantity	Start from	SI base units
force	F = ma	kg m s⁻²
energy	E = Fd	kg m² s⁻²
pressure	p = F / A	kg m⁻¹ s⁻²

These examples use relations taught later in the course. At this point, a question may supply the relation; focus on replacing each quantity with its base units.

Learn the usual named units, but do not guess their base-unit form. Start from a defining equation, substitute base units and simplify.

Worked example

Base units of power

Definition: $\text{power} = \dfrac{\text{work}}{\text{time}}$ .
Work has the units of energy: $\text{kg m}^2\,\text{s}^{-2}$ .
Divide by time: $\dfrac{\text{kg m}^2\,\text{s}^{-2}}{\text{s}} = \text{kg m}^2\,\text{s}^{-3}$ .

Answer

\text{power} = \text{kg m}^2\,\text{s}^{-3}

(9702/22/F/M/24 Q1(b))

Your turn9702/13/O/N/21 Q2 · 1 mark · adapted

Express the ohm in SI base units. Use resistance = potential difference ÷ current and potential difference = power ÷ current.

Show worked answer

First the volt: $[V] = \dfrac{\text{kg m}^2\,\text{s}^{-3}}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}$ .
Then the ohm: $[R] = \dfrac{[V]}{\text{A}} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}$ .

Answer

\text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-2}

Checking an equation: homogeneity

A physical equation is when every term has the same base units. A term is one complete part separated by $+$ , $-$ or the equals sign.
If even one term has different units, the equation is wrong. Units cannot be added to or equated with a different kind of quantity.
Matching units cannot prove an equation is right. A pure number like $\tfrac12$ or $2\pi$ has no units, so this check cannot see it.
In $s = vt$ , $s$ is displacement, $v$ is velocity and $t$ is time. The left side has unit m and the right side reduces to the same unit:

\text{m s}^{-1} \times \text{s} = \text{m}

The units match, so the equation passes the homogeneity check. This does not prove that the numerical factor or physical model is correct.

Worked example

Testing an equation for homogeneity

A mass hangs on a spring with spring constant $k$ . Test whether $T = 2\pi\sqrt{m/k}$ is homogeneous.

Find the base units of $k$ from $F = kx$ : $[k] = \dfrac{\text{kg m s}^{-2}}{\text{m}} = \text{kg s}^{-2}$ .
Then $\left[\dfrac{m}{k}\right] = \dfrac{\text{kg}}{\text{kg s}^{-2}} = \text{s}^2$ .
The square root gives $\sqrt{\text{s}^2} = \text{s}$ , which matches $T$ . The $2\pi$ has no units, so the equation is homogeneous.

Answer

\text{homogeneous}

(9702/12/F/M/24 Q3)

Worked example

One change: find the unknown power

$T = 2\pi(L/g)^{n}$ is homogeneous. Find $n$ .

$\left[\dfrac{L}{g}\right] = \dfrac{\text{m}}{\text{m s}^{-2}} = \text{s}^2$ .
Right side: $(\text{s}^2)^{n} = \text{s}^{2n}$ . Left side: $\text{s}$ .
Match the powers: $2n = 1$ , so $n = \tfrac{1}{2}$ .

Answer

n = \tfrac{1}{2}

(9702/12/M/J/25 Q4)

Worked example

One more change: base units of a constant

The power output of a star is $P = \sigma A T^4$ , where $A$ is area and $T$ is temperature. Find the base units of the constant $\sigma$ .

Make the constant the subject: $\sigma = \dfrac{P}{A T^4}$ .
Substitute base units:
$[P]=\text{kg m}^2\,\text{s}^{-3}$
$[AT^4]=\text{m}^2\,\text{K}^4$
$[\sigma]=\text{kg s}^{-3}\,\text{K}^{-4}$

Answer

[\sigma] = \text{kg s}^{-3}\,\text{K}^{-4}

(9702/13/O/N/24 Q3)

These three examples are the three ways the exam asks it: test an equation, find an unknown power, or find the base units of a constant. All three use the same skill: they replace each symbol with its base units, then compare.

Common mistake

Do not give units to a pure-number constant (

2\pi

and

4\pi^2

have none). And when one side has several terms, reduce every term, not just the first one.