Two identical open organ pipes are sounded together at 27^ C , and they produce a fundamental frequency of 300 ~Hz . If the temperature of the gas in one of the pipes is raised to 31^ C while the other remains at 27^ C , the number of beats heard per second will be:

Step-by-step solution

The fundamental frequency of an organ pipe is proportional to the square root of the absolute temperature of the gas inside it ( f T ). Initial temperature, T₁ = 27^ C = 300 ~K Final temperature of the heated pipe, T₂ = 31^ C = 304 ~K The new frequency f₂ is given by: f₂ f₁ = T₂ T₁ f₂ = 300 304 300 f₂ = 300 (1 + 4 300 )^ 1/2 Since the change in temperature is small, we can use the binomial approximation (1 + x)^ n 1 + nx for x 1 : f₂ 300 (1 + 1 2 4 300 ) f₂ 300 (1 + 2 300 ) = 300 + 2 = 302 ~Hz The beat frequency is