A guitar string is 80 cm long and has a fundamental frequency of 400 Hz. In its fundamental mode the maximum displacement is 2 cm at the middle. If the tension in the string is 10 6 dynes, what is the maximum of that component of the force on the end support which is perpendicular to the equilibrium position of the string?
Text Solution
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Use Cartesian coordinates with the x-axis along the equilibrium position of the string and the origin at one of its fixed ends. Then the two fixed ends are at x = 0 and x = l = 80 cm, as shown in Fig. At x = 0, the y-component of the force on the support is
F y = T sin θ 
Tθ 
T 
,
where T is the tension in the string. The guitar string has a sinusoidal form
y = y 0 sin 
with ω = 
= 
= 
, y 0 = 2 cm. Thus
y = 2 sin 
cm.
Hence at x = 0,
F y = – 
cos(ωt)
And
F y max = 
= 7.85 × 10 4 dynes.
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