Two uniform stretched wires A and B made of steel are vibrating under the same tension. If the first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B, the ratio olf the lengths of the strings is?

Dear Student ,
n is the number of overtone .
From the relation of the stretching of the of an wire with frequency and mass per unit length is given in the third step .
Here m is the mass per unit length .

The frequency is calculated using the equation,

 

 

L = resonating length.

 

T = tension at the string.

 

m = mass/unit length of the wire.

 

Derivation of the equation:

 

Laws of transverse vibrations of stretched strings;

 

 

 

 

 

 

Law of length:

 

For a given string under constant tension, the frequency of vibration is inversely proportional to the length of the string.

 

 

Law of tension:

 

For a given string of constant length, the frequency of vibration is directly proportional to the square root of the tension.

 

 

Law of mass:

 

For a string of constant length and under a constant tension, the frequency of vibration is inversely proportional to the square root of its mass per unit length. If M is the mass and L is the length of the string then;

 

 

 

 

If d is the diameter of the wire then;

 

 

 

Substituting in equation (1) we get;

 

 

The law of mass may be put into two additional laws, for strings of circular cross-section, as given below.

 


Hope this helps you .

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