A stretched string vibrates with frequency f, when tension in the string is T, for what value of tension, the frequency of the same string is doubled?

Explanation

The frequency of a stretched string is given by the formula f = (1/2L)√(T/μ), where T is the tension, L is the length of the string, and μ is the linear mass density. To double the frequency (2f), we set up the equation: 2f = (1/2L)√(T'/μ), where T' is the new tension. By squaring both sides and solving for T', we find that (2f)² = (1/2L)²(T'/μ). This leads to the conclusion that T' = 4T. Therefore, to double the frequency, the tension must be quadrupled. The other options are incorrect because:2T: Simply doubling the tension does not suffice to double the frequency, as frequency increases with the square root of tension.8T: Increasing to 8T would increase the frequency by a factor of 2.83, which is more than double.16T: Quadrupling the tension results in a frequency increase of 4 times, not just double.