A U-tube having unequal arm-lengths has water in it. A tuning ford of frequency 440 Hz can set up the air in the shorter arm in its fundamental mode of vibration and the same tuning fork can set up the air in the longer arm in its first overtone vibration. Find the length of the air columns. Neglect any end effect and assume that the speed of sound in air = 330 m s-1.

Sol. Given, m = 10 g = 10 × 10^–3 kg, l = 30 cm = 0.3 m Let the tension in the string will be = T μ = mass / unit length = 33 × 10^–3 kg The fundamental frequency ⇒ n0 = 1/2l √T/μ …(1) The fundamental frequency of closed pipe ⇒n base 0 = (v/4l) 340/4 * 50 * 10^2 = 170 Hzz …(2) According equation (1) * (2) we get 170 = 1/2 * 30 * 10^-2 * √T/33 * 10^-3 ⇒ T = 347 Newton.