Question

Explain why velocity increases when water flowing in a broad pipe enters a narrower pipe. A sonometer wire, 36 cm long, vibrates with a frequency of 288 Hz in the fundamental mode when it is under a tension of 24.5 N. Calculate the linear density of the material of the wire

Answer 1

A pipe narrows such that the same volume takes up more length, as the figure schematically illustrates. The cross-sectional area of the larger pipe is A₁, and the smaller pipe's is A₂. According to the continuity equation, v₂ = (A₁/A₂)v₁.

Since A₁/A₂ > 1, v₂ > v₁. For the same volume to pass points 1 and 2 in a given time, the speed must be greater at point 2.

The procedure can be reversed precisely. When the pipe widens, the fluid's speed drops if it is flowing in the other direction.

Data: L = 0.36 m, n = 288 Hz, T = 24.5 N

n = `1/(2L)  sqrt (T/m)`

∴ `m = T/ (2nL)^2`

= `24.5/((2 xx 288 xx 0.36)^2)`

= 5.699 × 10⁻⁴ kg/m