Figure shows different standing wave patterns on a string of linear mass density `4.0 xx 10^-2 kg//m` under a tension of 100 N. The amplitude of antinodes is indicated in each figure. The length of the string is 2.0m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii)Write down the transverse displacement y as a function of x and t for each mode. (Take the initial configuration of the wire in each mode to be as shown by the dark lines in the figure).

Answer

A, B, C, D

Explanation

(i) `f_(a) = v/(2l) = sqrt(T//mu)/(2l)`
`=sqrt((100//4xx(10^(-2))/(2xx2))`
`= 12.5 Hz. `
`f_b = 3(v/(2l)) = 3 fa = 37.5 Hz.
(ii) `omega = 2pi f`
`:. Omega_a = 2pif_a = 25 pi `
`omega_(b) = 2pif_(b) = 75pi`
`v = sqrt (T/mu) = sqrt(100/(4xx 10^(-2)`
`= 50 m//s`
`k = omega/v `
`:. k_a = (omega_a)/v = (25pi)/50 = pi/2`
`k_b = (omega_b)/v = (75pi)/50 = (3pi)/2 `