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Question
Explain how vibrating strings can be verified using a sonometer.
Solution
(1) Verification of law of length: According to this law, `"n" ∝ 1/"L"` if T and m are constant. The sonometer wire of given linear density m is kept under constant tension T to validate this law. The length of the wire is adjusted for the wire to vibrate in unison with tuning forks of different frequencies by the corresponding resonating lengths of the wire.
A set of tuning forks having different frequencies n1, n2, n3, ... are used and corresponding vibrating lengths of wire are noted as L1, L2, L3 ... by keeping the tension constant (T). We will observe that n1L1 = n2L2 = n3L3 = .... = constant, for the constant value of tension (T) and mass per unit length (m).
∴ nL = constant
i.e., `"n" ∝ 1/"L"`, if T and m are constant. Thus, the first law of a vibrating string is verified by using a sonometer.
(2) Verification of the law of tension: The vibrating length (l) of the given wire of mass per unit length (m) is kept constant for verification of the second law. By changing the tension the same length is made to vibrate in unison with different tuning forks of various frequencies. If tensions T1, T2, T3, ... correspond to frequencies n1, n2, n3, .... etc. It is found that, within experimental errors, `"n"_1/sqrt("T"_1)="n"_2/sqrt("T"_2)="n"_3/sqrt("T"_3)=....... = "constant"`
`or "n"/sqrt"T" = "constant"`
∴ `"n" ∝ sqrt"T"` if l and m are constant. This is the second law of a vibrating string.
(3) Verification of linear density: For verification of the third law of a vibrating string, two wires having different masses per unit length m1 and m2 (linear densities) are used. The first wire is subjected to suitable tension and made to vibrate in unison with the given tuning fork. The vibrating length is noted as (L1). Using the same fork, the second wire is made to vibrate under the same tension and the vibrating length (L2) is determined. Thus the frequency of vibration of the two wires is kept the same under the same applied tension T. It is found that,
`"L"_1sqrt("m"_1) = "L"_2sqrt"m"_2`
`"L"sqrt("m")` = constant
But by the first law of a vibrating string, `"n" ∝ 1/"L"` Therefore we get that, `"n" ∝ 1/sqrt("m")`, if T and L are constant. This is the third law of vibrating string.
RELATED QUESTIONS
State the laws of vibrating strings
Two wires of the same material and the same cross-section are stretched on a sonometer. One wire is loaded with 1.5 kg and another is loaded with 6 kg. The vibrating length of the first wire is 60 cm and its fundamental frequency of vibration is the same as that of the second wire. Calculate the vibrating length of the other wire.
A sonometer wire is stretched by the tension of 40 N. It vibrates in unison with a tuning fork of frequency 384 Hz. How many numbers of beats get produced in two seconds if the tension in the wire is decreased by 1.24 N?
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Explain the construction and working of sonometer hence explain its use to verify first law of vibrating string (Law of length).
