Dimensions of Physical Quantities - Units & Measurements Video Lecture - Class 11

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unit and measurement part 1 - part 9
before going ahead with part 10 we will
now start another interesting topic on
dimensions of physical quantities what
is dimension dimension of a physical
quantity shows how the quantity is
related to the seven basic quantities in
the previous slides I have told you that
there are two kinds of physical
quantities one basic quantities the
other is derived quantities so the basic
quantities in the basic quantities we
had major kilogram second ampere Kelvin
Angela mol these were the basic units of
the basic quantities so dimension of a
physical quantity even sure how this
physical quantity is derived from these
basic quantities because any other unit
will be derived from these basic units
so dimension of any physical quantity
will show that so how do we denote a
dimensional formula it is denoted in
terms of M L T etc enclosed in a square
braces for example M is used to denote
mass L for length T for time a for
current that is MP
and so on so let us take some example
for example we take this physical
quantity density we know that density is
equal to mass Phi for you so how do we
get the dimension for density now mass
is written as the dimension for mass
would be M this is the dimensional
formula for mass now volume is basically
length Hulme so we can write its
dimensional formula as n to the power 3
therefore density we can write as M
divided by L to the power 3 we can write
it as M L to the power minus 3 so this
would become the dimensional formula of
density so this formula shows how
density is derived from them basic
quantities these are the basic
quantities we will see few more examples
of dimensional formula let us suppose
you are asked to find the dimensional
formula for acceleration we know that
acceleration is equal to velocity per
unit time now velocity in term is equal
to displacement per unit time so this
becomes displacement per unit time
square now displacement will be denoted
in dimensional formula by n and time
will be denoted by T so this will be T
Square
therefore the dimensional formula for
acceleration would become L T to the
power minus 2 let us see another example
work how do we define both
well it is nothing but force multiply it
by displacement how do we define force
force is equal to mass into acceleration
now mass will be written as M
acceleration as we got it from here
would be written as n T to the power
minus 2 displacement will be written as
n therefore the dimensional formula
would become M L to the power 2 T to the
power minus 2 let's take yet another
example try to find out the dimensional
formula for pressure
what is pressure pressure is force per
unit area now what is force forces again
mass into acceleration what is area area
is nothing but length square so we can
write it as M for acceleration we have n
T to the power minus 2 divided by L
square so this comes out to be M n to
the power minus 1 T to the power minus 2
so this is the dimensional formula for
pressure so now I hope you understand
how to get a dimensional borneman up for
any physical quantity first of all you
try to break the physical quantity into
the very basic quantities then it will
become easier for you to write the
dimensional formula for the basic
quantities and then get the dimensions
for the physical quantity now what do we
mean by dimensional equation it is
nothing but equating a physical quantity
to its dimensional formula first you
write down the dimensional formula for
the quantity and then you just equate it
to the quantity that becomes the
dimensional equation for example
the fight one note that for work the
dimensional formula is this so when we
write W is equal to this this becomes
the dimensional equation similarly for
pressure when we write P is equal to
this it becomes the dimensional liquid
so the conclusion of dimensional
analysis is that a dimensionally correct
equation need not be correct actually or
need not be an exact equation but a
dimensionally incorrect equation must be
wrong so if an equation is dimensionally
incorrect that means that equation can
never exist but if an equation is
dimensionally correct it might be that
that equation is not correct because
sometimes there are some constants also
involved in an equation so dimensionally
incorrect equation is always in right
dimensionally correct equation is not
always right deducing relation among
physical quantities let us take another
example to show the application of
dimension analysis we all know that the
time period of a simple pendulum depends
on the length of the pendulum L and the
acceleration due to gravity G now if I
ask you to deduce the expression for T
with the help of dimension analysis how
would you do that let us consider that
the time period t is equal to some
constant k the length of the pendulum to
the power E and the G to the power P
because we already know that this T
depends on N and G all they do not know
is what is the power of dependence now
what we will do is we will equate the
dimensional formula of both LHS and RHS
so n
yes the dimensional formula routine this
or we can write it as M to the power 0 n
to the power 0 D to the power 1 in RHS
the dimensional formula would be n to
the power a G is acceleration due to
gravity so the dimensional formula for G
would be the same as acceleration
children in T to the power minus 2 to
the power D so we can write this as n to
the power a this will be n to the power
D T to the power minus 2 D or we can
write it as M to the power 0 L to the
power a plus B T to the power minus 2 me
now we equate LHS and RHS and equating
LHS and RHS the powers of L want to be
with power of L in NHS the 0 and power
of L in RHS is a plus P so we get a is
equal to minus P now we equate powers of
T so powers of T in LHS is 1 and in RHS
is minus to me therefore B is equal to
minus 1 by 2 now putting this value of
the in this equation we get a is equal
to 1 by 2 therefore the value of T will
become K L to the power 1 by 2 into G to
the power minus 1 by 2 that is equal to
K root over L by G so we could
the relation from this equation now this
can be considered as a constant which
cannot be derived dimensionally let us
now study dimensional and non
dimensional constants dimensional
constants there are certain quantities
which have dimensions but they have a
constant value for example speed of
light speed of light is nothing but
speed so speed of light will have the
dimension similar to that of speed that
is L T to the power minus 1 but it will
always have a constant Y it always has a
constant value evolve 3 into 10 to the
power 8 meter per second
similarly the gravitational constant
capital G it always has a constant value
but at the same time it has a
dimensional formula how do we get the
Tom dimensional coordinate for Capital Z
using Newton's universal law of
gravitation from Newton's law of
gravitation we know F is equal to G M 1
M 2 by r square or we can write G is
equal to F R square by M 1 M G now we
know that for F we can write force as
mass into acceleration R square is
nothing but length square divided by
mass into mass this mass and mass will
get cancelled so acceleration is L T to
the power minus 2 again length square is
L square and then M divided by n so the
dimensional coordinate um ZUP to be M to
the power minus 1 l to the
three and T to the power minus two
however is T all these have a constant
value so there are certain constants
which even though they have a
dimensional photograph but their values
are always constant then we will see the
non-dimensional constants there are
certain constant quantities these values
are constant and they do not have a
dimension for example Phi Phi has a
constant value of 22 by 7 and it doesn't
have any dimension at all thank you
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