Question

Hello can some one show and exlpain to me the derivation of this formula?

V = (Ft(mL^-1)^-1)^(2^-1)

or simply

V = sqrt (F/(m/L))

were

V = the veloctiy of a piece of string

Ft = the force of tension

m = mass

L = length

Thank you!

Answer 1

This is the velocity of a transverse wave on the string.
One way is to look at a length dx of the string at a changing angle to the line between end points.
At x, point 1, you have Force F horizontal to the left, F1y down and the resultant of those two along the string
At x+dx, point 2, you still have F horizontal but now to the right, F2y up, and the resultant of those two along the string.
The resultants must be along the string so at the left:
F1y/F = -dy/dx at x
and at the right
F2y/F = dy/dx at x+dx
Then the net force is in the y direction and is
F[ dy/dx at x+dfx - dy/dx at x]
which is the very definition of
F dx d^2F/dx^2
and must equal the mass times the acceleration
do
F dx d^2y/dx^2 = (m/L)dx d^2y/dt^2
or
d^2y/dx^2 = (m/LF)d^2y/dt^2
That is the wave equation with v^2 = L F/m
or velocity = sqrt (Tension/mass per unit length)