The velocity v, of a wave in a stretched string, depends on the tension T, in the spring and the mass per unit length of the spring. Obtain an expression fo...
Question 1 Report
The velocity v, of a wave in a stretched string, depends on the tension T, in the spring and the mass per unit length of the spring. Obtain an expression for v in terms of T and u, using the method of dimensions.
Answer Details
Dimensional derivation of the speed of a wave on a stretched string.
Assume the speed v depends on the tension T and the mass per unit length \(\mu\):
\[v=k\,T^{a}\mu^{b}\]
where k is a dimensionless constant. Writing the dimensions:
- \([v]=LT^{-1}\)
- \([T]=MLT^{-2}\) (force)
- \([\mu]=ML^{-1}\) (mass per unit length)
So \(LT^{-1}=(MLT^{-2})^{a}(ML^{-1})^{b}=M^{a+b}L^{a-b}T^{-2a}\).
Equating powers:
- M: \(a+b=0\)
- L: \(a-b=1\)
- T: \(-2a=-1\Rightarrow a=\tfrac{1}{2}\)
From these, \(b=-\tfrac{1}{2}\). Therefore
\[v=k\,T^{1/2}\mu^{-1/2}=k\sqrt{\dfrac{T}{\mu}}.\]
Thus the speed of the wave is proportional to the square root of the tension divided by the mass per unit length.
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