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 = PV2 = ML -1T-2L6 =[ML5T-2]

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

= [L3]

ANS a = [ML5 T-2]

b = [L3]

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: loge x or log10 x

(b) Exponential functions: ex, ax

(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 = Fr2 = M L T-2 L2 = [M L3 T-2]

No because e-kr is dimension less

So k = 1/r

= [L-1]

Ans: dimensional formula of

C = [M L3T-2]

K = [L-1]