Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously?

Question

Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously? (A) 0.01 (B) 0.02 (C) 0.03 (D) 0.04

Tardigrade Admin · Accepted Answer

Beats per second when both the wires vibrate simultaneously,

n_{1} \pm n_{2} = 6

Or

\frac{1}{2 i} \frac{T}{m} \pm \frac{1}{2 l} \frac{T ^{'}}{m} = 6

Or

\frac{1}{2 l} \frac{T ^{'}}{m} - \frac{1}{2 l} \frac{T}{m} = 6

Or

\frac{1}{2 l} \frac{T ^{'}}{m} - 600 = 6

\frac{1}{2 l} \frac{T ^{'}}{m} = 606

Given that fundamental frequency

\frac{1}{2 l} \frac{T}{m} = 600

Dividing Eq.(i) by Eq. (ii), we get

\frac{\frac{1}{2 l} \frac{T ^{'}}{m}}{\frac{1}{2 l} \frac{T}{m}} = \frac{606}{600}

Or

\frac{T ^{'}}{T} = 1.01

Or

\frac{T ^{'}}{T} = 1.02%

Or

T^{'} = T 1.02

Increase in tension

Δ T^{'} = T \times 1.02 - T

= 0.02 T

Hence,

Δ T^{'} = 0.02

Q. Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously?

0.01

0.02

0.03

0.04

Q. Two identical piano wires have a fundamental frequency of $600$ cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of $6$ beats per second when both wires vibrate simultaneously?