Dimensional Analysis Making new formula - 3

Description:

Frequency of a vibrating string (v) depends on length of string ( l ), Tension in string ( T ) and Linear mass density ( m ).

Solution

To form a new formula we will use dimensional analysis.

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

v = kl^a T^b m^c

Where,

v = Frequency

k = Arbitrary constant

l = Length of string

T = Tension in string and

m = Linear mass density.

Step 2 − Writing dimensions of each quantity.

Frequency  = [T^-1]

Tension (force) = [M L T^-2]

Linear mass density = [M L^-1]

Substituting the values −

M⁰ L⁰ T^-1 = [L]^a [M L T^-2]^b [M L^-1]^c

M⁰ L⁰ T^-1 = [M]^{b + c} [L]^{a + b -c} [T]^-2b

Step 3 − Compare similar dimensions.

After comparing similar dimensions, we get −

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

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

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

v = kl^-1 T^1/2 m^-1/2

v = k/l √T/m