A sonometer string vibrates with a frequency of 400 Hz . When the length of the string is halved and the tension is altered, it begins to vibrate with a frequency of 200 Hz . The ratio of the new tension to the original tension in the string is:

Step-by-step solution

The frequency of a vibrating string is given by the formula f = 1 2L T , where L is the length, T is the tension, and is the linear mass density of the string. Let the initial frequency be f₁ = 400 Hz, length be L₁ = L , and tension be T₁ = T . Then f₁ = 1 2L T . Let the final frequency be f₂ = 200 Hz, length be L₂ = L/2 , and tension be T₂ = T' . Then f₂ = 1 2(L/2) T' = 1 L T' . Taking the ratio of the two frequencies: f₂ f₁ = 1 L T' 1 2L T = 2 T' T . Substituting the given values: 200 400 = 2 T' T 1 2 = 2 T' T T'