A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform strin

Step 1: The fundamental frequency of a closed pipe is given by $f = frac{v}{4L}$, where $v$ is t

A hollow pipe of length 0.8 m is closed at one end. At its...

A hollow pipe of length $0.8 \mathrm{~m}$ is closed at one end. At its open end a $0.5 \mathrm{~m}$ long uniform string is vibrating in its second harmonic and it resonates with the fundamentals frequency of the pipe if the tension in the wire is $50 \mathrm{~N}$ and the speed of sound is $320 \mathrm{~ms}^{-1}$, what is the mass of the string. (A) $20 \mathrm{~g}$ (B) $5 \mathrm{~g}$ (C) $40 \mathrm{~g}$ (D) $10 \mathrm{~g}$

A hollow pipe of length $0.8 \mathrm{~m}$ is closed at one end. At its open end a $0.5 \mathrm{~m}$ long uniform string is vibrating in its second harmonic and it resonates with the fundamentals frequency of the pipe if the tension in the wire is $50 \mathrm{~N}$ and the speed of sound is $320 \mathrm{~ms}^{-1}$, what is the mass of the string.
(A) $20 \mathrm{~g}$
(B) $5 \mathrm{~g}$
(C) $40 \mathrm{~g}$
(D) $10 \mathrm{~g}$

Key Concepts

Wave Speed on a Stretched String

The speed at which waves propagate along a stretched string is determined by the tension in the string and its linear mass density. This speed can be calculated using the formula v = ?(T/?), where T is the tension force and ? (mu) is the mass per unit length, illustrating how both physical parameters affect the wave propagation characteristics.

Relationship between Tension, Linear Mass Density, and Wave Speed

This relationship is fundamental in the study of waves on a string. It connects the mechanical properties of the string, such as how taut it is (tension) and its mass distribution (linear mass density), to the speed at which disturbances travel along the string. This concept is crucial for determining properties like frequency when the wave speed and length of the string are known.

Resonance in Closed Pipes

In acoustics, a closed-end pipe supports standing waves with a node at the closed end and an antinode at the open end. The fundamental frequency of such a pipe is given by the speed of sound divided by four times the length of the pipe, which represents the lowest resonant frequency that can be sustained in that system.

Harmonic Frequencies in Vibrating Strings

Vibrating strings fixed at both ends support standing waves whose frequencies are integer multiples of the fundamental frequency. The n-th harmonic has a frequency that is n times the fundamental frequency, with the second harmonic being twice the frequency of the fundamental mode. This concept is essential in matching the resonance conditions between two different vibrating systems.

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