Question

Write an expression for the speed of transverse waves in string. Prove its corrections by the method of dimensional analysis.

Answer 1

Speed of Transverse waves in a string:- The speed of transverse waves in a stretched string is given by,

where T is the tension (or force) in the string and μ is the mass per unit length of the string. μ is also called the linear density of the string. Its unit in S.I. system is kg m^-1.

The correctness of the relation (i) can be checked by the method of dimensional analysis. Dimensional formula of

v = [M⁰L¹T^-1]

Dimensional formula of T

T = Force = [M¹L¹T^-2]

Dimensional formula of

μ = [M¹L^-1T⁰]

From equation (i)

Since dimensional formula of R.H.S. is same as that of L.H.S. So, the relation (i) correctly represents the speed of transverse wave along a string.

Eqn. (i) can also be expressed as:

If 'r' be the radius of the string and ρ be the density of the material of the string, then

μ = mass of the string per unit length

= Volume of string per unit length x density

= πr² x 1 x ρ = πr²ρ

Hence, equation (i) can be written as

v = [T/πr²ρ]   ...(ii)

It is clear from equation (ii) that speed of the transverse wave in a string is inversely proportional to the square root of the density of the material of the string, i.e., v ∝ 1/√ρ. That is why speed of the waves in the string made of iron is less than in the string made of aluminium, as density of iron is more than that of aluminium.