Fundamental, Derived and Supplementary Quantities and Units with Dimensions

Fundamental (Base) Quantities and Units

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Fundamental quantities are quantities that are independent of one another and cannot be expressed in terms of other quantities. Their units are termed fundamental or base units.

A fundamental quantity (or base quantity) is defined as a physical quantity that does not depend on any other quantity, and thus cannot be defined or expressed in terms of any other physical quantities. They act as the core foundation from which all derived quantities (such as speed, force, energy, or volume) are calculated by either multiplying or dividing the base units.

The Système International d'Unités (or the International System of Units) S.I units has seven fundamental quantities.:

The fundamental quantities, their S.I units and symbols are shown via the table below.

Quantity	SI unit	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric current	Ampere	A
Thermodynamic Temperature	Kelvin	K
Amount of Substance	Mole	mol
Luminous intensity	Candela	cd

 

Below are useful prefixes regarding the symbols of S.I units:

• Kilo (k) = 10³

• Mega (M) = 10⁶

• Giga (G) = 10⁹

• Centi (c) = 10⁻²

• Milli (m) = 10⁻³

• Micro (μ) = 10⁻⁶

• Nano (n) = 10⁻⁹

• Pico (p) = 10⁻¹²

Please read on the measurement of basic quantities and importance of measurement in physics here.

 

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Derived Quantities and Units

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Derived quantities are obtained by multiplying or dividing fundamental quantities. For this reason, their units are "derived units"

A derived quantity is defined as a physical quantity that comes from combining two or more base (fundamental) quantities through multiplication or division. You cannot measure them directly; instead, you calculate them using mathematical formulas.

The table below show the derived quantities, how they are obtained (formula), derived units and based units.

Quantity	Formula	Derived unit	Base units
Area	Length × Width (l × w)	m²	m²
Volume	Length × Breadth × Height (l × b × h)	m³	m³
Density	Mass/Volume (m/V)	kg/m³	kg m⁻³
Velocity	Displacement/Time (s/t)	m/s	m s⁻¹
Acceleration	Change in Velocity/Time (Δv/t)	m/s²	m s⁻²
Momentum	Mass × Velocity (m × v)	kg m/s	kg m s⁻¹
Force	Mass × Acceleration	Newton (N)	kg m s⁻²
Pressure	Force/Area (F/A)	Pascal (Pa)	kg m⁻¹ s⁻²
Work/Energy	Force × Distance  (F × d)	Joule (J)	kg m² s⁻²
Power	Work/Time (W/t)	Watt (W)	kg m² s⁻³
Frequency	1/Time Period	Hertz (Hz)	s⁻¹
Electric charge	Current × Time (I × t)	Coulomb (C)	A s
Potential difference	Work done/Quantity of electricity (W/Q)	Volt (V)	kg m² s⁻³ A⁻¹
Resistance	Potential difference/Current (V/I)	Ohm (Ω)	kg m² s⁻³ A⁻²

 

With regards to the above table, consider the Example below:

The unit of force, 1 N = 1 kg m s⁻². This implies that one newton is the force that gives a 1 kg mass an acceleration of 1 m/s².

Please read the introduction to physics, its definition, branches and importance here.

 

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Supplementary Quantities and Units

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These are two dimensionless quantities dealing with angles.

Supplementary quantity in physics is defined as a dimensionless physical quantities used to measure angles, consisting of the plane angle and the solid angle.

• Plane Angle (θ): It is defined as the ratio of arc length (l) to radius (r), and its unit is the radian (rad). It measures the angle formed in a two dimensional plane.

• Solid Angle (Ω): It measures the three-dimensional "cone" or angular spread subtended at the center of a sphere. It is defined as the ratio of the intercepted surface area (A) to the square of the radius (r²), and its unit is the steradian (sr).

The table below shows the supplementary quantities, their unit, symbol and definition.

Quantity	Unit	Symbol	Definition
Plane angle	radian	rad	Angle at the centre of a circle subtended by an arc equal in length to the radius (θ = arc/radius)
Solid angle	steradian	sr	Solid angle at the centre of a sphere subtended by an area on the surface equal to r² (Ω = area/r²)

 

Note the foollwing:

• 2π rad = 360°, so 1 rad ≈ 57.3

• A full sphere subtends 4π steradians.

• Supplementary quantities have no dimensions because they are ratios of two lengths (or areas),

Note: In 1995 the SI reclassified them as dimensionless derived units. Many syllabuses still list them as supplementary, so know both.

You can read on the basic and fundamental physics concepts here.

 

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Dimensional Analysis

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Dimension of a physical quantity is the way it is related to the fundamental quantities. These dimensions are denoted by square brackets [].

The table shows the fundamental quantities and their dimensions.

Fundamental Quantity	Dimension
Length	[L]
Mass	[M]
Time	[T]
Electric current	[I]
Temperature	[K1] or [Θ]
Amount of Substance	[N]

 

The dimension of a physical quantity refers to the powers (or exponents) to which the base (fundamental) quantities must be raised to represent that specific quantity.

Dimensional formula shows us which base quantities and powers make up (build) a derived quantity.

The table below shows us the dimensions of common derived quantities.

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Dimensions of Common Quantities:

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Quantity	Dimension
Area	L²
Volume	L³
Density	ML⁻³
Velocity	LT⁻¹
Acceleration	LT⁻²
Momentum	MLT⁻¹
Force	MLT⁻²
Pressure	ML⁻¹T⁻²
Work, Energy	ML²T⁻²
Power	ML²T⁻³
Frequency	T⁻¹
Electric Charge	IT
Angle, Strain, Refractive Index	dimensionless

You can read an introduction to velocity and acceleration here.

 

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How to find a Dimension

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This is done by writing the formula, and replacing each quantity by its dimension. Next, we simplify. An example is illustrated below:

• Pressure = force/area

• Pressure = MLT⁻² / L²

• Pressure = ML⁻¹T⁻²

Finding the dimensions of an unknown quantity in a physical equation relies directly on the principle of homogeneity.

 

Principle of Homogeneity

In a correct physical equation, every term must have the same dimensions on both sides. Only quantities with the same dimensions can be added or subtracted.

The principle of homogeneity of dimensions states that the dimensions of every term on both sides of a physically correct equation must be identical.

You can read on the density of various substances here.

 

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Applications of Dimensional Analysis

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1. Checking whether an equation is dimensionally correct. Below are examples:

• Test the formula below.
• s = ut + ½at²

[s] = L

[ut] = (LT⁻¹)(T)

(LT⁻¹)(T) = L

[ut] = L

[½at²] = (LT⁻²)(T²)

(LT⁻²)(T²) = L

[½at²] = L (the ½ is a number with no dimension)

Therefore, [s] = L, [ut]= L and [½at²] = L (All terms are L, so the equation is dimensionally consistent).

 

• Test the formula below:
• v = u + at²

[v] = LT⁻¹

[u] = LT⁻¹

[at²] = LT⁻²·T²

LT⁻²·T² = L

[at²] = L

[v] = LT⁻¹, [u] = LT⁻¹ and  [at²] = L (All terms are not L, so the equation is dimensionally inconsistent).

 

2. Deriving the dimension or unit of a quantity. Below is an example.

• Find the dimensions and unit of G in:
• F = Gm₁m₂/r²

G = Fr²/(m₁m₂)

G = (MLT⁻²)(L²)/M²

(MLT⁻²)(L²)/M² = M⁻¹L³T⁻²

G = M⁻¹L³T⁻² with unit Nm² kg⁻² (or m³ kg⁻¹ s⁻²).

 

3. Converting from one system of units to another. Below is an example.

• Convert 1 g/cm³ to kg/m³:*

1 g/cm³ = (10⁻³ kg)/(10⁻⁶ m³)

(10⁻³ kg)/(10⁻⁶ m³) = 1000 kg/m³

 

4. Deriving a formula (relationship between quantities). Below is an example.

• The period T of a simple pendulum depends on its mass m, length l and g. Find the formula.

Step 1: Write the relation as an equation

Let T = k mᵃ lᵇ gᶜ, where k is a dimensionless constant.

Step 2:  Substitute the fundamental dimensions: The dimensional formulas for each of the quantities are:

• Period (T): [M⁰ L⁰ T¹]

• Mass (m): [M¹ L⁰ T⁰)

• Length (l): [M⁰ L¹ T⁰]

• Acceleration due to gravity (g): [M⁰ L¹ T^(-2)]

Substituting these into our initial equation:: Period (T) = [M⁰ L⁰ T¹]

Dimensions: T = Mᵃ · Lᵇ · (LT⁻²)ᶜ

Mᵃ · Lᵇ · (LT⁻²)ᶜ = Mᵃ  L^(b+c)  T^-2c

Step 3: Equate the exponents of M, L and T. By comparing the powers on both sides, we get a system of linear equations:

For M: a = 0

For T: -2c = 1, c = -1/2

For L: b + c = 0

b + (-)1/2 = 0

b = 1/2

Step 4: Substitute the exponents back into the equation

Now substitute(a = 0, b = 1/2, and c = --1/2 into the original equation:

T= k . m⁰ . l^1/2 . g^1/2

Compare powers: M gives a = 0. T gives −2c = 1, so c = −½. L gives b + c = 0, so b = ½.

Conclusion

The dimensional formula for the period T itself is simply [M⁰ L⁰ T¹].

The derived physical relationship shows that the period does not depend on mass (m⁰) and is given by T = k√(l/g) (where experimental data proves k = 2π

Therefore, T = k√(l/g). The mass does not matter. Experiment or theory gives k = 2π.

You can read on the concept of heat and temperature here.

 

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Limitations of Dimensional Analysis

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1. It cannot determine dimensionless constants (like ½, 2π, k).

2. IIt cannot distinguish quantities with the same dimensions (e.g. work and torque, or speed and velocity).

3. It does not work for trigonometric, logarithmic or exponential functions of a quantity, because their arguments must be dimensionless.

4. It cannot derive an equation with more than one term added together (e.g. s = ut + ½at²) unless additional information is used.

5. A dimensionally correct equation is not necessarily physically correct. Dimensions test consistency, not truth.

6. It requires prior knowledge of which quantities are involved.

Please read on the next topic (position, distance and displacement) here.

 

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Practice Questions

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Section A: Objective Questions

1. Which of these is a fundamental quantity? A. Force B. Mass C. Velocity D. Energy

2. The SI unit of luminous intensity is _____. A. lumen B. lux C. candela D. watt

3. Which of the following is not a derived quantity? A. Density B. Area C. Time D. Volume

4. The dimension of pressure is _____. A. MLT⁻² B. ML⁻¹T⁻² C. ML²T⁻² D. ML⁻³

5. The unit of power in base SI units is _____. A. kg m² s⁻² B. kg m s⁻² C. kg m² s⁻³ D. kg m⁻¹ s⁻²

6. The least count of a vernier callipers in centimetres is usually _____. A. 0.1 B. 0.01 C. 0.001 D. 1.0

7. A micrometer screw gauge has a pitch of 0.5 mm and 50 thimble divisions. Its least count is _____. A. 0.5 mm B. 0.1 mm C. 0.05 mm D. 0.01 mm

8. An ammeter is connected _____. A. in parallel with the component B. in series with the component C. across the battery only D. either way

9. A body of mass 6 kg is on Earth where g = 10 m/s². Its weight is _____. A. 0.6 N B. 6 N C. 16 N D. 60 N

10. The dimension of frequency is _____. A. T B. T⁻¹ C. LT⁻¹ D. MT⁻¹

11. Which of these is not an SI base unit? A. kelvin B. mole C. newton D. ampere

12. Which equation is dimensionally correct? A. v = u + at² B. s = ut + ½at² C. v² = u + 2as D. s = ut + at

13. The unit "joule" in base units is _____. A. kg m s⁻² B. kg m² s⁻² C. kg m² s⁻³ D. kg m s⁻¹

14. The radian is a _____. A. fundamental unit with dimension L B. derived unit with dimension LT⁻¹ C. supplementary (dimensionless) unit D. unit of solid angle

15. A vernier callipers has a zero error of +0.03 cm. The observed reading is 4.52 cm. The correct reading is _____. A. 4.55 cm B. 4.49 cm C. 4.52 cm D. 4.50 cm

 

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Section B: Theory

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1. (a) Define measurement. (b) State five importances of measurement.

2. Distinguish between fundamental and derived quantities, giving three examples of each with their SI units.

3. List the seven fundamental quantities and their SI units and symbols.

4. State four differences between mass and weight. Name the instrument for measuring each.

5. Describe how to read a vernier callipers. What is zero error, and how is it corrected?

6. A micrometer screw gauge reads 5.5 mm on the sleeve and 28 on the thimble. Its zero error is −0.02 mm. Find the correct reading of the diameter.

7. Describe how you would find the period of a simple pendulum accurately using a stopwatch. Why time many oscillations?

8. What are supplementary quantities? Define the radian and the steradian.

9. Define dimension. Determine the dimensions of (a) work, (b) power, (c) the spring constant k in F = kx, (d) charge.

10. State the principle of homogeneity of dimensions.

 

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Calculations and Dimensional Analysis

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1. Check the dimensional correctness of the following:

   (a) F = mv²/r

   (b) P = ½ρv² (pressure in a moving fluid)

   (c) v = √(gh)

   (d) E = mc²

2. The speed v of a wave on a stretched string is v = √(T/μ), where T is tension and μ is mass per unit length. Show that the equation is dimensionally consistent.

3. The frequency f of a vibrating string depends on its length l, tension T and mass per unit length μ. Use dimensional analysis to show that f = k(1/l)√(T/μ).

4. Convert (a) 72 km/h to m/s, (b) 5 g/cm³ to kg/m³, (c) 2.5 N to kg cm s⁻².

5. Find the dimensions and SI unit of the coefficient of viscosity η in F = 6πηrv, where r is radius and v is velocity.

6. The centripetal force is thought to depend on mass m, speed v and radius r. Use dimensional analysis to find the relationship.

7. A student claims the period of a pendulum is T = 2π√(g/l). Use dimensions to show whether it is correct.

8. Explain why dimensional analysis cannot tell you the value of the constant 2π in T = 2π√(l/g).