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प्रश्न
State the laws of vibrating strings
State and explain the laws of vibrating strings.
उत्तर
The fundamental frequency of a vibrating string under tension is given as, n = `1/(2l)sqrt("T"/"m")`
From this formula, three laws of vibrating string can be given as follows:
- Law of length: The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant.
∴ n α `1/l` ...........….(if T and m are constant.) - Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of tension if the vibrating length and mass per unit length are constant.
∴ n α `sqrt"T"` ......….(if l and m are constant.) - Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density), if the tension and vibrating length of the string are constant.
∴ n α `1/sqrt"m"` ..........….(if T and l are constant.)
If r is the radius and r is the density of the material of string, linear density is given as
Linear density = mass per unit length
= volume per unit length × density
= `(pir^2l)/l xx rho`
= πr2ρ
As n ∝ `1/sqrt"m"`, if T and l are constant, we have,
n ∝ `1/sqrt(pir^2rho)`
i.e., n ∝ `1/sqrt(rho)` and n ∝ `1/"r"`
Thus the fundamental frequency of vibrations of a stretched string is inversely proportional to the radius of string and the square root of the density of the material of vibrating string.
संबंधित प्रश्न
Explain how vibrating strings can be verified using a sonometer.
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?
Two wires of the same material and same cross-section are stretched on a sonometer. One wire is loaded with 1 kg and another is loaded with 9 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.
When tension T is applied to a sonometer wire of length l, it vibrates with the fundamental frequency n. Keeping the experimental setup the same, when the tension is increased by 8 N, the fundamental frequency becomes three times the earlier fundamental frequency n. The initial tension applied to the wire (in newton) was ______.
A stretched wire of length 260 cm is set into vibrations. It is divided into three segments whose frequencies are in the ratio 2 : 3 : 4. Their lengths must be ______.
A segment of wire vibrates with a fundamental frequency of 450 Hz under a tension of 9 kg weight. The tension at which the fundamental frequency of the same wire becomes 900 Hz is ____________.
A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and the corresponding resonating length is L1· If the weight is reduced to `("w"/4)`, the corresponding resonating length becomes L2. The ratio `("L"_1/"L"_2)` is ______.
A pipe closed at one end has length 0.8 cm. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 320 m/s, the mass of the string is ______.
Two identical strings of length 'l' and '2l' vibrate with fundamental frequencies 'N' hertz and '1.5 N' hertz, respectively. The ratio of tensions for smaller length to larger ______.
The fundamental frequency of a string stretched with a weight 'M' kg is 'n' hertz. Keeping the vibrating length constant, the weight required to produce its octave is ______.
In the sonometer experiment, the string of length 'L' under tension vibrates in the second overtone between the two bridges. The amplitude of vibration is maximum at ______
A sonometer wire of length 'ℓ1' is in resonance with a frequency of 250 Hz. If the length of the wire is increased to 'ℓ2', then 2 beats per second are heard. The ratio of lengths `ℓ_1/ℓ_2` of wire will be ______
When a sonometer wire vibrates in the third overtone, there are ______.
Two metal wires 'P' and 'Q' of same length and material are stretched by same load. Their masses are in the ratio m1:m2. The ratio of elongation of wire 'P' to that of 'Q' is ______.
Explain the construction and working of sonometer hence explain its use to verify first law of vibrating string (Law of length).
