Question

A student stands several metres 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 75.0 cm long, has a mass of 2.25 g, and is under a tension of 400 N. A second student, moving towards the wall, hears 8.30 beats per second. Determine:

a. frequency of sound produced by the wire;

b. the speed at which the second student approaches the wall.

Answer 1

When the string is vibrating in the third harmonic then thelength of the string is

                    L = 3λ / 2

                    λ = 2L / 3 = 2* 0.75 /3 =0.5 m

Formula for the speed (v) of the waves on the string is =

  

= sqrt( 400 / (2.25*10^-3/0.75)) = 365.148 m/s

Now frequency (f) of the string is f = v / λ = 365.148 /0.5 =730.29600 Hz answer

Speed of source is vs = 0
Speed (v) of the sound in air = 340m/s
Now when the observer is approaching the stationary observerthen the formula for the apparent frequency heard by the observeris
f' = [(v + vo) / v]f
= [1 + (vo / v)]f
= f + (vo / v)f
f' - f = (vo / v)f
Here f' - f = 8.3 beats per second

f = 730.296Hz
v = 340m/s

so

8.3 = Vo /340 *730.296

Vo = 3.86 m/s answer

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