Checking Correctness of Formulas

Description:

Dimensional analysis has the following steps −

Step 1 − Write the relation with assumed powers and an arbitrary constant.

Step 2 − Writing dimensions of each quantity.

Step 3 − Compare similar dimensions.

Step 4 − Substitute the power of dimensions in the equation formed in step 1.

Example

Time of oscillation of pendulum depends upon length of pendulum, mass of bob and acceleration due to gravity.

To form a new formula we will use dimensional analysis.

Step 1 − Write the relation with assumed powers and an arbitrary constant.

T α m^a l^b g^c

T = k m^a l^b g^c

Where,

T = Time of oscillation

k = Arbitrary constant

m = Mass of bob

l = Length of pendulum and

g = Acceleration due to gravity

Step 2 − Writing dimensions of each quantity.

M⁰ L⁰ T¹ = [M]^a [L]^b [L T^-2]^c

M⁰ L⁰ T¹ = M^a L^b L^c T^-2c

M⁰ L⁰ T¹ = [M]^a [M]^a [L]^b+c [T]^-2c

Step 3 − Compare similar dimensions.

After comparing similar dimensions, we get −

a = 0,b + c = 0, -2c = 1

a = 0,b = 1/2, c = -1/2

Step 4 − Substitute the power of dimensions in the equation formed in step 1.

T = kl ^1/2 g^-1/2

T = k√l/g