Question

9. Show dimensionally that the frequency n of transverse waves in a string of
length I and mass per unit length m under a tension T is given by n =k/l√T/M​

Answer 1

Answer:

Well, it has been given to us experimentally that the frequency of the transverse waves on the string can potentially depend upon the length mass per unit length and tension

So,

Dimensionally for frequency on the left hand side we want 1/T = T^-1 on the LHS and the same on the RHS

Dimension of Tension(same as force) = MLT^-2

Dimension of Mass per unit length = ML^-1

And,

dimension of Length = L

Let, RHS be such that

= [MLT^-2]^a * [L]^b * [ML^-1]^c

= M^a+c * L^a+b-c * T^-2a

Now,

on left hand side exponents of M and L are zero and of T is -1

So,

clearly

-2a = -1

or,

a = 1/2

and so from the equations

c = -1/2

and,

b = -1

So,

the formula for the frequency comes out to be

Frequency = Length^-1 * Tension^1/2 * Mass per unit length^-1/2

Or,

n = 1/l * √T/m