Question

A 75 g bungee cord has an equilibrium length of 1.2 m. The cord is stretched to a length of 1.8 m, then vibrated at 20 Hz. This produces a standing wave with two antinodes. What is the spring constant of the bungee cord (in N/m)?

Answer 1

The standing wave with two antinodes must be for the 1st overtone of a line clamped at each end, and so with a node in the centre. Meaning that the wavelength of the standing wave is 1.8m.

The wave speed is given by V = frequency x wavelength
V = 20 . 1.8 = 36 m/s

The wave speed can also be calculated from
V = SqRt(Tension / Mass per unit length)

36 = SqRt (T / (0.075/1.8) )

Rearrange 36^2 x 0.075 = 1.8 x T => T = 54 N

Last step... F = k . x
x = extension, F = extending force
k = spring constant - the answer.
F = the Tension! Hurray

56 = k ( 1.8 - 1.2 ) => k = 90 N/m