The frequency of the note emitted by a stretched string is 300 Hz. What is the frequency of this note when:
(a) Length of the wave is reduced by one third without changing tension.
(b) The tension is increased by one-third without changing the length of the wire.
(Ans: 450 Hz, 346 Hz)
Data Given:
Frequency of note = f = 300 Hz
To Find:
(a) Frequency of note = f₁ = ? {When length of the wave is reduced by one third without changing tension.}
(b) Frequency of note = f₂ = ? {When the tension is increased by one-third without changing the length of the wire. }
Solution:
(a) Frequency of note = f₁ = ? {When length of the wave is reduced by one third without changing tension.}
According to given conditions let original wavelength λ is reduced by 1313 then the new wavelength λ₁ will be
λ₁ = λ - λ3 = 2λ3
As given there is no change occur in the tension F then the speed will remains constant.
v₁ = v
or
f₁ λ₁ = f λ
or
f₁ = fλλ₁
putting value of λ₁ = 2λ3
f₁ = fλ2λ3
or
f₁ = 3f2
putting the value of f = 300 Hz (given)
f₁ = 3x300Hz2
f₁ = 450 Hz -----------------Ans.
Hence the new frequency will increase to 450 Hz
(b) Frequency of note = f₂ = ? {When the tension is increased by one-third without changing the length of the wire. }
According to given conditions let the tension F is increased by 13 then the new tension F₂ will be
F₂ = F + F3 = 4F3
Formula for fundamental frequency
f = 13L √Fm ----------(1)
and the new frequency after increasing tension F₂ = 4F3
f₂ = 13L √F₂m
or
f₂ = 13L √4F3m -------(2)
Now dividing equation (2) by (1) we will get
or
f₂ = 1.155 x f
putting f = 300 Hz (given)
f₂ = 1.1547 x 300 Hz
f₂ = 346.41 Hz
or
f₂ = 346 Hz -----------------Ans. (2)
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