Question

The equation of vibration of a stretched string fixed at both ends and vibrating in 5th harmonic is $y=3sin(0.4x)cos(20 \pi t)$, where x and y are in cm and t in second. Length of the string is

• A. 12.5 $\pi$ cm
• B. 8.5 $\pi$ cm
• C. 10.5 $\pi$ cm
• D. 4.5 $\pi$ cm

Answer 1

Answer: (A)

12.5 $\pi$ cm

Answer 2

The string vibrates in 5th harmonic. For a string fixed at both ends the relation is

$kL = n\pi$

In the given equation

$y = 3 sin(0.4x) cos(20\pi t)$,

the spatial part gives k = 0.4 rad/cm and n = 5. Thus,

$L = (n\pi)/k = (5\pi)/(0.4) = 12.5\pi$ cm.