Homogeneity

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: log_e x or log₁₀ 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 = 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]