Question

two indentical sonometer wires have a fundamental frequency of 1000hz when kept under the same tension. What fractional increase in the tension of one wire would cause an occurrence of 5beats/sec, when both wires vibrate together?

Answer 1

The fractional increase in the tension of one wire to cause an occurrence of the beat frequency is 1/100.

How to calculate the tension of the wires?

The formula for the frequency and tension of a vibrating wire is calculated as follows.

`f = (1)/(2L) \sqrt{(T)/(\mu) }`

where;

L is the length of wire

T is the tension of the wire

μ is mass per unit length

Keep every parameters constant, the relationship between frequency and tension is;

f ∝ √T

`(f_1)/(√(T_1) ) = (f_2)/(√(T_2) ) \\\\√(T_2) = (f_2 √(T_1) )/(f_1)`

The formula beat frequency is given as;

Fb = |F₂ - F₁|

5 Hz = F₂ - 1000 Hz

F₂ = 1005 Hz

The fraction increase in the tension of one wire is;

`√(T_2) = (f_2 √(T_1) )/(f_1)\\\\√(T_2) = (1005 √(T_1) )/(1000)\\\\√(T_2) = 1.005 √(T_1) \\\\√(T_2) = √(1.01T_1)\\\\ T_2 = 1.01T_1\\\\T_2 = (1 + 0.01)T_1\\\\`

T₂ = (100% + 1%)T₁

T₂ = (101%)T₁

So fraction increase in tension of one wire is 1 %.