state the law of vibrating strings
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Laws of vibrating strings
First law
The fundamental frequency of vibration of a stretched string is inversely proportional to its length when its tension and linear density are constant.
Second law
the fundamental frequency of vibration of a stretched string is directly proportional to the square root of the tension when the length and linear density are constant.
Third law
the fundamental frequency of the vibration of stretched string is inversely proportional to the square root of linear density when the length and tension are constant.
For the verification of all the above three laws a sonometer is used.
Description of a sonometer
Sonometer is used for measuring the intensity of the sound through vibrating strings. The sonometer consists of a hollow rectangular wooden box of length 1m. The box contains few holes on the sides for the free vibrations of air inside. A wire is fixed at end, which passes over a frictionless pulley and other end is attached with a weight hanger. By changing the weight, the tension in the wire can be changed. There are two movable bridges B1 and B2 over which wire passes and by changing the distance between them, the length l of the vibrating segment of the wire can be changed. A meter scale is fixed on the board to know the resonating length.
Principle of a sonometer
A small paper rider of inverted “V" shape is placed on the stretched wire between B1 and B2. When a tuning fork is excited and is placed on the wooden box by the side of the wire, and if the fundamental frequency of the stretched wire and that of the tuning fork are equal, then resonance takes place. The wire vibrates with maximum amplitude and paper rider falls off.
Verification of first law
the string of the sonometer is kept under suitable tension by attaching a certain load to the weight hander. The two bridges B1 and B2 are kept very close and paper rider is placed on the wire between the two bridges. A tuning fork of known frequency 'n' is excited and placed on the wooden box. The distance between the two bridges is increased gradually until the paper rider flies off. In this position, the fundamental frequency of the vibrating segment of the wire is equal to frequency of the tuning fork. The two are in resonance and the distance between the two bridges is noted as resonating length 'l'. Keeping the tension constant and with the same wire the experiment is repeated with tuning forks of different frequencies and corresponding resonating lengths are noted. The obtained results satisfies the condition of first law. Hence the first law is verified.
Verification of second law
It is difficult to verify second law directly. The second law can be verified indirectly by showing square root of tension divided by length as constant. A load is attached to the weight of hanger and the tension “T" of the wire is noted. The two bridges are kept very closed and paper rider is placed at the middle. A tuning fork is excited and placed on the box beside the wire. The distance between the bridges is increased gradually until the paper rider flies off. The resonating length is noted as 'l'. The experiment is repeated with the same wire and same tuning fork for different tensions and corresponding resonating lengths are noted. By showing square root (tension T)/(Length (l) = constant. Hence the second law is verified.
Verification of third law
It is very difficult to verify third law directly. It can be verified indirectly by showing l*square root (m)=constant.
A wire of linear density 'm' is connected to the board and a suitable tension is applied. The two bridges are kept very close and the paper rider is placed at middle. A turning fork is excited and placed on the wooden box. The distance between the two bridges is gradually increased length is noted as 'l'. The experience is repeated with the same tension and same tuning fork but with different wires of different linear densities and the corresponding resonating lengths are noted. And the product of length and square root of density of each reading are taken and checked. We get the required proof. Hence third law is also verified.
Precautions
1) The pulley must be smooth and friction less.
2) Paper rider must be placed on the wire exactly at center between the bridges.
3) The thickness of the wire should be uniform.
4) Weight hanger should not touch the experimental bench
5) Tension applied should be with in the elastic limit of the wire.
6) The turning fork must always be held at its stem only
7) The tuning fork is placed on wooden box such that the prongs of it may not touch the wire.
Uses of Sonometer
1) To determine the velocity of transverse waves along a stretched string.
2) To measure alternating current frequency in a circuit
3) To determine the unknown frequency of the tuning fork
4) To verify the laws of transverse waves along a stretched string.