Page 45 - Class-11-Physics-Part-1_Neat

UNITS AND MEASUREMENT 31

For example, 12.9 g – 7.06 g, both specified to three square brackets [ ]. Thus, length has the
significant figures, cannot properly be evaluated dimension [L], mass [M], time [T], electric current
as 5.84 g but only as 5.8 g, as uncertainties in [A], thermodynamic temperature [K], luminous
subtraction or addition combine in a different intensity [cd], and amount of substance [mol].
fashion (smallest number of decimal places rather The dimensions of a physical quantity are the
than the number of significant figures in any of powers (or exponents) to which the base
the number added or subtracted). quantities are raised to represent that
quantity. Note that using the square brackets
(3) The relative error of a value of number
specified to significant figures depends not [ ] round a quantity means that we are dealing
only on n but also on the number itself. with ‘the dimensions of’ the quantity.
In mechanics, all the physical quantities can
For example, the accuracy in measurement of be written in terms of the dimensions [L], [M]
mass 1.02 g is ± 0.01 g whereas another and [T]. For example, the volume occupied by
measurement 9.89 g is also accurate to ± 0.01 g.
The relative error in 1.02 g is an object is expressed as the product of length,
= (± 0.01/1.02) × 100 % breadth and height, or three lengths. Hence the
3
3
= ± 1% dimensions of volume are [L] × [L] × [L] = [L] = [L ].
Similarly, the relative error in 9.89 g is As the volume is independent of mass and time,
= (± 0.01/9.89) × 100 % it is said to possess zero dimension in mass [M°],
= ± 0.1 % zero dimension in time [T°] and three
Finally, remember that intermediate results in dimensions in length.
Similarly, force, as the product of mass and
a multi-step computation should be acceleration, can be expressed as
calculated to one more significant figure in Force = mass × acceleration
every measurement than the number of = mass × (length)/(time) 2
digits in the least precise measurement.
2
These should be justified by the data and then The dimensions of force are [M] [L]/[T] =
–2
the arithmetic operations may be carried out; [M L T ]. Thus, the force has one dimension in
otherwise rounding errors can build up. For mass, one dimension in length, and –2
example, the reciprocal of 9.58, calculated (after dimensions in time. The dimensions in all other
rounding off) to the same number of significant base quantities are zero.
figures (three) is 0.104, but the reciprocal of Note that in this type of representation, the
0.104 calculated to three significant figures is magnitudes are not considered. It is the quality
9.62. However, if we had written 1/9.58 = 0.1044 of the type of the physical quantity that enters.
Thus, a change in velocity, initial velocity,
and then taken the reciprocal to three significant average velocity, final velocity, and speed are
figures, we would have retrieved the original all equivalent in this context. Since all these
value of 9.58. quantities can be expressed as length/time,
This example justifies the idea to retain one their dimensions are [L]/[T] or [L T ].
–1
more extra digit (than the number of digits in
the least precise measurement) in intermediate 2.9 DIMENSIONAL FORMULAE AND
steps of the complex multi-step calculations in DIMENSIONAL EQUATIONS
order to avoid additional errors in the process The expression which shows how and which of
of rounding off the numbers. the base quantities represent the dimensions
of a physical quantity is called the dimensional
2.8 DIMENSIONS OF PHYSICAL QUANTITIES formula of the given physical quantity. For
example, the dimensional formula of the volume
The nature of a physical quantity is described is [M° L T°], and that of speed or velocity is
3
by its dimensions. All the physical quantities [M° L T ]. Similarly, [M° L T ] is the dimensional
–2
-1
represented by derived units can be expressed formula of acceleration and [M L –3 T°] that of
in terms of some combination of seven mass density.
fundamental or base quantities. We shall call An equation obtained by equating a physical
these base quantities as the seven dimensions quantity with its dimensional formula is called
of the physical world, which are denoted with the dimensional equation of the physical

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