# Physics 9702 Doubts | Help Page 180

__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 = m

^{2}
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 mv

^{2}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 = mv

^{2}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 / m

^{3}= 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 / m

^{2}. 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}m^{3 }= m^{3}s^{-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^{-1}m^{3}= 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}m^{1}s^{1}) = m^{-1}s
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^{-1}s^{-2}
Thus, the equivalent of Pascal (Pa)
in SI units is

**kgm**.^{-1}s^{-2}
4/O/N/02 Q.2(b),Q.3(b)(i),Q.5(b),Q.6(c)(i)

ReplyDeleteFor 4/O/N/02 Q.2(b), see solution 888 at

Deletehttp://physics-ref.blogspot.com/2015/07/physics-9702-doubts-help-page-181.html