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Question 885: [Measurement > Units > Fundamental and Derived units]

Explain fundamental and derived units.

Solution 885:

There are base (fundamental) SI units for describing the 7 base quantities. These are:

• mass [unit = kilogram (kg)],
• length [unit = metre (m)],
• time [second (s)],
• current [ampere (A)],
• temperature [kelvin (K)],
• amount of substance [mole (mol)] and
• luminous intensity [candela (cd or cdl)].

From these 7 base units, other units can be obtained by performing different operations (multiplication or division) with the base units to describe derived quantities. These other units are called derived units.

Examples:

Area = length × length. Units of area = m × m = m²

Speed = length / time. Units of speed = m / s = ms^-1

Now, the units of some derived quantities are given special names.

Examples:

(1)

Force = mass × acceleration. Units of force = kg ms^-2.

The unit of force is called Newton (N). So, the equivalence of a Newton in SI base units is kgms^-2.

(2)

Frequency = 1 / time. Units of frequency = 1 / s = s^-1. The unit is also called Hertz.

In this way, the units for more complex quantities can be derived in terms of base units.

Question 886: [Measurement > Units > Homogeneity]

(a) How would you check the homogeneity of an equation?

(b) An obstacle of width a is placed in a fluid having a density ρ, and viscosity η. The streamline flow of the fluid becomes turbulent if the flow speed exceeds a critical value v. Which of the equations below relates v, a, ρ and η correctly?

(i) v = Aηa / ρ                          (ii) v = Bη / aρ                                     (iii) v = Cρa / η

Solution 886:

(a)

The homogeneity of an equation means that the units of both sides of the equation should be the same. This can be checked by first writing the units of the components / quantities involved into SI base units. The overall (simplified) units on both sides of the equation should be the same.

An equation which is not homogeneous is definitely incorrect, while a homogeneous equation may not always be correct (there may be a numerical value [which has no units] missing in certain cases).

Example:

Kinetic energy = ½ mv² is correct since the overall units on the right-hand side is the same as the units of energy which is joule, when written in SI unit (you can work this on your own).

However, Kinetic energy = mv² is incorrect even if the units is the same as in the above case.

(b)

The equation which relates these quantities should be homogeneous (should have same overall base units on both sides of the equation). To check the homogeneity, the base units of each quantities involved should be known.

Assuming that A, B and C are constants with no units,

Units of speed v = ms^-1

Units of density ρ = [mass] / [volume] = kg / m³ = kgm^-3

Unit of width a = m

{I don’t think the units of viscosity should be known at AS level. It should be given. Anyway, the units of viscosity is Pascal (Pa) × second (s). 1 Pa = [Force] / [Area] = N / m². So, units of viscosity η = N s m^-2 = (kg ms^-2) s m^-2 = kg m^-1 s^-1. }

Since the speed (having units: ms^-1) is given on one side of the equation, the overall unit on the other side should be ms^-1 for the equation to be homogeneous.

Consider (i) v = Aηa / ρ

{Since A, B and C are assumed to be constants with no units,}

Units of ηa / ρ = (kg m^-1 s^-1) m / (kgm^-3) = (kg m^-1 s^-1) m kg^-1 m³ = m³s^-1

So, equation (i) is not homogeneous.

Consider (ii) v = Bη / aρ

Units of η / aρ = (kg m^-1 s^-1) / [m (kgm^-3)] = (kg m^-1 s^-1) m^-1 kg^-1m³ = ms^-1

Equation (ii) is homogeneous since the same units are obtained on both sides.

Consider (iii) v = Cρa / η

Units of ρa / η = (kgm^-3) m / (kg m^-1 s^-1) = (kgm^-3) m (kg^-1 m¹ s¹) = m^-1s

Equation (iii) is also incorrect since it is inhomogeneous.

This question could also be changed, giving the correct equation, and asking to find the units of viscosity. In such a case, make viscosity the subject of formula and reduce the units of all the other quantities (which are known) to an overall unit.

Question 887: [Measurement > Units]

Pressure P is due to a liquid of density ρ is related to depth h by the expression

P = ρgh

where g is acceleration of free fall. Use this expression to determine derive units of pressure. Explain your working.

Solution 887:

This question is similar to June 2009 Paper 22 Question 1(b).

Density ρ = mass / volume

Units of density ρ: kg m^-3

Units of acceleration g: m s^-2

Units of pressure P: kgm^-3 ms^-2 m = kgm^-1s^-2

Thus, the equivalent of Pascal (Pa) in SI units is kgm^-1s^-2.