Question

a student stands several meters in front of a smooth reflecting wall, holding a board on which a wire is fixed at each end. the wire, vibrating in its third harmonic, is 75.0cm long, has a mass of 2.25g, and is under a tension of 400 N. a second student, moving towards the wall, hears 8.30 beats per second. what is the speed of the student approaching the wall? (solve without calculus)

Answer 1

Step-by-step explanation:

From the question we are told that

The length of the wire is
`L = 75.0cm = (75)/(100) = 0.75 \ m`

The mass of the wire is
`m = 2.25 \ g = (2.25)/(1000) = 0.00225 \ kg`

The tension is
`T = 400 \ N`

The frequency of the beat heard by the second student is

`f_b = 8.30\ beat/second`

The speed of the wave generated by the vibration of the wire is mathematically represented as

`v = \sqrt{(TL)/(m)}`

substituting values

`v = \sqrt{(400 *0.75)/(0.00225)}`

`v = 365.15 m/s`

The wire is vibrating in its third harmonics so the wavelength is

`\lambda = (2L)/(3)`

substituting values

`\lambda = (2*0.75)/(3)`

`\lambda = 0.5 \ m`

The frequency of this vibration is mathematically represented as

`f = (v)/(\lambda )`

substituting values

`f = (365.15)/(0.5 )`

`f = 730.3 Hz`

The speed of the second student (Observer) is mathematically represented as

`v_o = [(f_b)/(2f) ] * v`

substituting values

`v_o = [(8.30)/(2* 730.3) ] * 365.15`

`v_o = 2.08 \ m/s`