**Law of homogeneity **

Here we can jump through +, - , = (signs)

and a,b, c- are Constant then find dimensional formula of a, b, c-

**Apply Law of homoginity**

Dimensional formula of v = dim-formula of at = v

Similarly we can say t = c Þ C = [T]

b = [L]

**Example: **The equation of state of some gases can be expressed as here p is pressure, V is the volume, T is temp. then the dimensional formula of Constants a and b will be ?

Apply Law of homoginity Dimensional formula of p = Dim-formula of

a = PV^{2} = ML ^{-1}T^{-2}L^{6} =[ML^{5}T^{-2}]

Similarly dimensional formula of v = Dim. Formula of b b = v

= [L^{3}]

**ANS** a = [ML^{5} T^{-2}]

b = [L^{3}]

**Application No-3 **

**Dimension less quantities or functions**

(i) Quantities having units but dimensionless

So all trigonometric functions will be dimension less i.e. sin *x*, cos *x*, tan *x* – etc.

(ii) Quantities neither having units nor dimension

Refraction index

Reynolds number, Relative density etc.

(iii) Dimension less mathematical functions

(a) Logarithmic functions: *log _{e}*

*x*or

*log*

_{10}

*x*

(b) Exponential functions: e^{x}, a^{x}

(a) P

(b) R

(c) T

(d) V

**Correct answer is (a)**

**Example** Nuclear force between nucleons is given by Here F is force, r is distance, and c, k are Constants then find dimensional formula of C and k = ?

Here e^{-kr} is dimension less

So C = Fr^{2} = M L T^{-2} L^{2} = [M L^{3} T^{-2}]

No because e^{-kr} is dimension less

So k = 1/r

= [L^{-1}]

**Ans:** dimensional formula of

C = [M L^{3}T^{-2}]

K = [L^{-1}]