A piano wire has length L and mass M. If its fundamental frequency is f, its tension is:

Question

i got T=(lambda*f)^2m/L, but their answer is 4LMF^2 and i cant figure out how to get that

Answer 1

the wavelength of the fundamental is twice the length of the wire.
L = n lambda/2 where n=1 for fundamental
L = lambda/2
but lambda = vT = v/f where v is speed
so
L = v/2f
now v = sqrt (TL/M)
so
2 f L = sqrt T L/M
4 f^2 L^2 = T L/M
T = 4 L M f^2