Units and Measurement n Flashcards
what is the nature of physical quantitiy
The nature of a physical quantity is described
by its dimensions. All the physical quantities
represented by derived units can be expressed
in terms of some combination of seven
fundamental or base quantities. We shall call
these base quantities as the seven dimensions
of the physical world, which are denoted with square brackets [ ]. Thus, length has the
dimension [L], mass [M], time [T], electric current
[A], thermodynamic temperature [K], luminous
intensity [cd], and amount of substance [mol].
what is the dimension of physical quantities?
The dimensions of a physical quantity are the
powers (or exponents) to which the base
quantities are raised to represent that
quantity.
in dimensions of physical quantities, we do not deal with magnitudes explain.
the
magnitudes are not considered. It is the quality
of the type of the physical quantity that enters.
Thus, a change in velocity, initial velocity,
average velocity, final velocity, and speed are
all equivalent in this context. Since all these
quantities can be expressed as length/time,
their dimensions are [L]/[T] or [L T–1].
what is the dimensional formula?
The expression which shows how and which of
the base quantities represent the dimensions
of a physical quantity is called the dimensional
formula of the given physical quantity. For
example, the dimensional formula of the volume
is [M° L3 T°]
what is the dimensional equation?
An equation obtained by equating a physical
quantity with its dimensional formula is called
the dimensional equation of the physical quantity. Thus, the dimensional equations are
the equations, which represent the dimensions
of a physical quantity in terms of the base
quantities.
[V] = [M0 L3 T0]
what are the applications of dimensional analysis
- The recognition of concepts of dimensions, which
guide the description of physical behaviour is
of basic importance as only those physical
quantities can be added or subtracted which
have the same dimensions.
-A thorough
understanding of dimensional analysis helps us
in deducing certain relations among different
physical quantities and checking the derivation,
accuracy and dimensional consistency or
homogeneity of various mathematical
expressions.
When magnitudes of two or more
physical quantities are multiplied, their units
should be treated in the same manner as
ordinary algebraic symbols. We can cancel
identical units in the numerator and
denominator.
The same is true for dimensions
of a physical quantity. Similarly, physical
quantities represented by symbols on both sides
of a mathematical equation must have the same
dimensions.
State the principle of homogeneity of dimensions
A physical equation is dimensionally correct only if the dimensions of the terms on both sides of the equations are the same.
i,e
A+B=C
implies
[A]=[B]=[C]
when can physical quantities be added/subtracted?
Magnitudes of physical quantities can be added or substracted only if they have same dimensions. Force cannot be added to time neither can electric current be subtracted from thermodynamic temperature.
Where is principleof homegeneity of dimensions used?
It is used to check the correctness of an equation.
Dimensions are customarily used as a
preliminary test of the consistency of an
equation, when there is some doubt about the
correctness of the equation. However, the
dimensional consistency does not guarantee
correct equations. If an equation fails this consistency
test, it is proved wrong, but if it passes, it is
not proved right. Thus, a dimensionally correct
equation need not be actually an exact
(correct) equation, but a dimensionally wrong
(incorrect) or inconsistent equation must be
wrong. This is because constant numbers are not taken into consideration.
why cant principle of homogeneity of dimensions gaurantee a correct equations
dimensional consistency does not guarantee
correct equations. It is uncertain to the extent
of dimensionless quantities or functions. The
arguments of special functions, such as the
trigonometric, logarithmic and exponential
functions must be dimensionless. A pure
number, ratio of similar physical quantities,
such as angle as the ratio (length/length),
refractive index as the ratio (speed of light in
vacuum/speed of light in medium) etc., has no
dimensions.
what is the 2nd application of dimensional analysis?
Dimensional analysis is very useful in deducing
relations among the interdependent physical
quantities. However, dimensionless constants
cannot be obtained by this method. The method
of dimensions can only test the dimensional
validity, but not the exact relationship between
physical quantities in any equation.
what are drawbacks of dimensional analysis
-> This method does not give information about the dimensionless constant K
-> It fails when a physical quantities depends on more than three physical quantities
-> It fails when a physical quantity is the sum or difference of two or more physical quantities
-> it fails to derive relationships which involve trigonometric, logarithmic or exponential functions.
-> it fails to distinguish between physical quantities that have the same dimensions
