Fundamental And Derived Measurement

Fundamental Measurement in physics

Note: terms used
Examples:
(1) ^ means raised to power

Physical Quantities

Physical quantities are quantities that have measurable physical properties. They consist of numerical value and unit and are classified into two: Fundamental physical quantities and Derived physical quantities.

Fundamental physical quantities

The fundamental physical quantities are basic qualities that provide the basic units of measurement.
Examples of three most important fundamental qualities are – length, mass and time.
The standard units of measurement of these physical quantities are metre (m), kilogram (kg) and second (s) respectively. By definition, top↑

Length, (l): is a measure of the distance between two separate and distinct points.
Mass, (m): measure of quantity of matter contained in an object.
Time, (t): is a measure of duration of an event.
Other related fundamental quantities are Electric current, luminous intensity, amount of substance and thermodynamic temperature.

Derived physical quantities

Derived physical quantities are quantities obtained by combining two or more fundamental quantities. That is, they are quantities that can be obtained by multiplying or by dividing two or more fundamental quantities. top↑

Examples

Derived
1)

velocity = Displacement
                  Time Taken
= L/t²

2)

Acceleration = change in velocity
                          Times taken
=L/t²

3)force     =     Mass × acceleration.
= m × L/t² = mL/t²
4)work      =     Force × distance.
mL/t² ×L = mL²/t²

Dimensions

The dimensions of a physical quantity indicates the way the fundamental quantities: Mass, Lenth and Time as represented by the letters M, L and T respectively, can be used to relate the fundamental units of a physical quantity.
For example, if velocity = displacement/Time
and displacement is the measure of change in length.
then, the dimension of velocity is L/T or LT-¹
top↑

Application of dimensions

Dimension is of great use in finding the true relationship between physical quantities, where the correct mathematical relation or formula cannot be easily obtained.  It also helps in determining the appropriate unit of a physical quantity.
Example, if in an experiment, a string fixed at both ends is plucked, it would be seen to vibrate. The velocity, v of the wave produce depends on the:
(i) tension, The force in the string
(ii) length, The lenth of the vibrating string and
(iii) its mass, m
This implies that the velocity of the wave produced is proportional to tension, length of the string and its mass, but the nature of proportionality of the velocity of the wave to the other physical quantities are not easily ascertained except by the use of dimension
Assuming
V = K Fa|bMc    ………..(1)
Where k constant of proportionality and a, b and c are numbers we are expected to find by dimension.
By dimension of velocity, (v), force, (F), length, (L) and the mass (m), the dimensions of the above physical quantities would be as follows:top↑
Velocity, V     =    L/T or LT-¹
Force, F          =    ML/T² or MLT-²
Length, l        =    L
Mass, m        =   M
Substituting the dimensions obtained in equation (1), it then becomes.

LT-¹      =   K(MLT-²)aLbMc
Equating the corresponding powers of M, L and T on both sides, it follows that for powers of
(i) M   :    0    =     a + c  ……. (a)
(ii) L   :    1    =     a + b  ……. (b)
(iii) T   :    -1   =    -2a   …….. (c)

Solving eqn (a), (b) and (c) simultaneously
Eqn (a)
a = ½
Eqn (b)
b = ½
Eqn (c)
c = -½
Equation (1) can now be expressed as
V = K1 F½ L½ M^-½

V = K √FL
           m

NOTE
Addition or subtraction of quantities does not affect dimension
Dimensions is only affect by multiplication and division. top↑

LET STOP HERE FOR TODAY……….

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