Answer to Question #102987 in Mechanics | Relativity for Promise Omiponle

Question

A 5.00 m, 0.732 kg wire is used to support two identical, uniform 235 N posts of equal lengths at an angle of 57.0 degrees above the horizontal. Assume the wire is horizontal. A strong wind is blowing, causing the wire to oscillate in its fundamental (first harmonic) mode.
a) What is the wavelength of the fundamental mode?
b) What is the wave speed of the transverse waves on the wire that are combining to yield the fundamental standing wave pattern that is observed? (Hint: The posts are in equilibrium; calculate the torque about the pivots of the beams to determine the tension in the wire.)
c) What would be the frequency of the fifth harmonic?

Answer 1

For the equilibrium:

$$(0.5L)mg\sin{(90-57)}=TL\sin{57}$$

Tension:

$$T=mg\frac{\sin {33}}{\sin{57}}=235\frac{\sin {33}}{2\sin{57}}=76\ N$$

a)

$$\lambda=2L=2(5)=10\ m$$

b)

$$v=\lambda f$$

$$f=\frac{1}{\lambda}\sqrt{\frac{TL}{m}}$$

$$f=\frac{1}{10}\sqrt{\frac{(76)(5)}{0.732}}=2.28\ Hz$$

c)

$$f_5=\frac{5}{10}\sqrt{\frac{(76)(5)}{0.732}}=11.4\ Hz$$