The method of dimensional analysis is used to
1. convert a physical quantity from one system of units to another.
2. check the dimensional correctness of a given equation.
3. establish a relationship between different physical quantities in an equation.

*Uses of dimensional
analysis*

The method of dimensional analysis is used to

1. convert a
physical quantity from one system of units to another.

2. check
the dimensional correctness of a given equation.

3. establish
a relationship between different physical quantities in an equation.

*(i) To convert a physical quantity from one
system of units to another*

Given the value of G in cgs system is 6.67 x 10^{âˆ'8}*dyne cm ^{2}*

**In cgs system**

G_{cgs} = 6.67
x 10^{âˆ'8}

M_{1} = 1g

L_{1} =
1 cm

T_{1} =
1s

**In SI system**

G = ?

M_{2} = 1 kg

L_{2} =
1m

T_{2} =
1s

The dimensional formula for gravitational
constant is [M^{-1}L^{3}T^{-2}]

In cgs system, dimensional formula for G is [M_{1}^{x}L_{1}^{y}T_{1}^{z}]

In SI system, dimensional formula for G is [M_{2}^{x}L_{2}^{y}T_{2}^{z}]

Here x = âˆ'1, y = 3, z = âˆ'2

[M_{2}^{x}L_{2}^{y}T_{2}^{z}]
= G_{cgs}[M_{1}^{x}L_{1}^{y}T_{1}^{z}]

G = G_{cgs}[M_{1}/m_{2}]^{x}[M_{1}/m_{2}]^{y}[M_{1}/m_{2}]^{z}

= 6.67 x 10^{âˆ'8} [1g/1kg]^{-1}[1cm/1m]^{3}[1s/1s]^{-2}

= 6.67 x 10^{âˆ'11}

In SI units,

G = 6.67 x 10^{âˆ'11} N m^{2} kg^{âˆ'2}

**(ii) To check the dimensional correctness of a
given equation**

Let us take the equation of motion

**s = ut + (****1/2****)at ^{2}**

Applying dimensions on both sides

[L] = [LT^{âˆ'1}] [T] + [LT^{âˆ'2}]
[T^{2}]

(1/2 is a constant having no dimension)

[L] = [L] + [L]

As the dimensions on both sides are the same,
the equation is dimensionally correct.

** **

**(iii) To establish a relationship between the
physical quantities in an equation**

Let us find an expression for the time period
T of a simple pendulum. The time period T may depend upon (i) mass m of
the bob (ii) length l of the pendulum and (iii) acceleration due to gravity g
at the place where the pendulum is suspended.

(i.e) T Î± m^{x} l^{y} g^{z}

T = k m^{x} l^{y} g^{z}

where k is a dimensionless constant of
propotionality. Rewriting equation (1) with dimensions,

[T^{1}] = [M^{x}] [L ^{y}]
[LT^{âˆ'2}]^{z}

[T^{1}] = [M^{x} L ^{y
+ z} T^{âˆ'2z}]

Comparing the powers of M, L and T on both
sides

x = 0, y + z = 0 and âˆ'2z = 1

Solving for x, y and z, x = 0, y = 1/2 and z =-1/2

From equation (1), T = k m^{o} l ^{1/2} g^{âˆ'}^{1/2}

T =k root(l/g)

Experimentally the value of k is determined to
be 2Ï€.

T=2Ï€ root(l/g)

*Limitations of
Dimensional Analysis*

(i) The
value of dimensionless constants cannot be determined by this method.

(ii) This
method cannot be applied to equations involving exponential and trigonometric
functions.

*(iii)* It
cannot be applied to an equation involving more than three physical quantities.

*(iv)* It
can check only whether a physical relation is dimensionally correct or not. It
cannot tell whether the relation is absolutely correct or not. For example
applying this technique *s =ut + (**1/4)** at ^{2} is dimensionally correct
whereas the correct relation is s = ut + (*

^{}

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