Question

Two taut strings of identical mass and length are stretched with their ends fixed, but the tension in one string is 1.10 times greater than in the other. Waves on the string with the lower tension propagate at 35.2 m/s. The fundamental frequency of that string is 258 Hz. What is the beat frequency when each string is vibrating at its fundamental frequency?

Answer 1

The beat frequency when each string is vibrating at its fundamental frequency is 12.6 Hz

Step-by-step explanation:

Given;

velocity of wave on the string with lower tension, v₁ = 35.2 m/s

the fundamental frequency of the string, F₁ = 258 Hz

velocity of wave on the string with greater tension;

`v_1 = \sqrt{(T_1)/(\mu )`

where;

v₁ is the velocity of wave on the string with lower tension

T₁ is tension on the string

μ is mass per unit length

`v_1 = \sqrt{(T_1)/(\mu) } \\\\v_1^2 = (T_1)/(\mu) \\\\\mu = (T_1)/(v_1^2) \\\\ (T_1)/(v_1^2) = (T_2)/(v_2^2)\\\\v_2^2 = (T_2v_1^2)/(T_1)`

Where;

T₁ lower tension

T₂ greater tension

v₁ velocity of wave in string with lower tension

v₂ velocity of wave in string with greater tension

From the given question;

T₂ = 1.1 T₁

`v_2^2 = (T_2v_1^2)/(T_1) \\\\v_2 = \sqrt{(T_2v_1^2)/(T_1)} \\\\v_2 = \sqrt{(1.1T_1*(35.2)^2)/(T_1)}\\\\v_2 = √(1.1(35.2)^2) = 36.92 \ m/s`

Fundamental frequency of wave on the string with greater tension;

`f = (v)/(2l) \\\\2l = (v)/(f) \\\\thus, (v_1)/(f_1) =(v_2)/(f_2) \\\\f_2 = (f_1v_2)/(v_1) \\\\f_2 =(258*36.92)/(35.2) \\\\f_2 = 270.6 \ Hz`

Beat frequency = F₂ - F₁

= 270.6 - 258

= 12.6 Hz

Therefore, the beat frequency when each string is vibrating at its fundamental frequency is 12.6 Hz