The phase speed | v | of a traveling transverse wave on a flexible string can pl

Question

The phase speed | v | of a traveling transverse wave on a flexible string can plausibly depend only on two quantities, the tension force | F_T | on the string (which tells you how strongly the string is pulled back toward equilibrium when it is disturbed) and the string's mass per unit length mu (which tells you about how long it takes the string to respond to a given restoring force). Assume that the speed depends on some product of powers of these quantities. Show that if these assumptions are true, dimensional analysis requires that | v | = C Squareroot | F_r |/mu where C is some unknown unitless constant. (It turns out that C = 1, as problem Q1D.1 shows.)

Explanation / Answer

Given that /v/ = C*sqrt(FT/mu)

FT is the tensional force ,units of FT are Newton

1 newton = kg-m/s^2

mu is the linear density ,units of mu are kg/m

and also given that C = 1 and also a unitless number

then v = C*sqrt(FT/mu)

units of v is m/sec

then

m/sec = 1*sqrt(kg-m/sec^2 /(kg/m))

m/sec = 1*sqrt(m^2/sec^2)

m/sec = m/sec

Hence proved the given equation is dimensionally equal

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