The equation for the vibration of a string fixed at both ends vibrating in its third harmonic is given by `y=2cm` `sin[(0.6cm^-1)x]cos[(500pis^-1)t]`. The length of the string is

Options

A. `15.7` cm
B. `20.6` cm
C. `24.6` cm
D. `12.5` cm

Answer

A

Explanation

For the third harmonic, the string of length L , vibrates with 3 loops.

`:. (3lambda)/2 = L " " :. lambda/2 = L/3 `
Comparing the equation `y = 2 sin ( 0.6x) cos ( 500 pi t) ` with the standard equation
`:. y = 2A cos ((2pi x)/lamda) sin ((2 pi t)/T) `
we get `(2 pi x)/lambda = 0.6 x :. (2 pi)/ lambda = 0.06 :. lambda = (2pi)/(0.6)`
` :. L = (3 lambda)/2 = 3/2 ((2pi)/(0.6)) = (3 pi)/(0.6)`
` = 5 pi = 5 xx 3.14 = 15.7 ` cm