Question

The frequency of vibration of a string may depend upon length ,tension and mass per unit length of the string.Use method of dimension for establishing the formula for frequency

Answer 1

Here, Bold T denotes tension and T denotes time

[n] = [T]^-1

[T] = [MLT^-2]

[l] = [L]

[m] = [ML^-1]

Now let us assume a relation

n = kT ^xl^ym^z

=> [ n] = k[T] ^x[l]^y[m]^z

=> [T]^-1 = k[MLT^-2]^x[L]^y[ML^-1]^z

=> [T]^-1 = k[M^x+z L^x+y-z T^-2x]

Comparing the powers on both sides of T

–2x = –1

=> x = ½

Comparing the powers on both sides of L

x + y – z = 0

=> z – y = ½

Comparing the powers on both sides of M

x + z = 0

=> z = –½

=> y = –1

So our expression becomes

n = kT ^xl^ym^z

=> n = kT^1/2 l^-1m^-1/2

=> n = (k/l)√(T/m)