A sonometer wire of density d and radius r is held between...

A sonometer wire of density $d$ and radius $r$ is held between two bridges at a distance $L$ apart. The wire has a tension $T$. The fundamental frequency of the wire will be (A) $f=\frac{1}{2 L r} \sqrt{\frac{T}{\pi d}}$ (B) $f=\frac{r}{2 L} \sqrt{\frac{\pi d}{T}}$ (C) $f=\frac{1}{2 L r} \sqrt{\frac{d}{\pi T}}$ (D) $f=\frac{1}{2 L} \sqrt{\frac{d}{T}}$

A sonometer wire of density $d$ and radius $r$ is held between two bridges at a distance $L$ apart. The wire has a tension $T$. The fundamental frequency of the wire will be
(A) $f=\frac{1}{2 L r} \sqrt{\frac{T}{\pi d}}$
(B) $f=\frac{r}{2 L} \sqrt{\frac{\pi d}{T}}$
(C) $f=\frac{1}{2 L r} \sqrt{\frac{d}{\pi T}}$
(D) $f=\frac{1}{2 L} \sqrt{\frac{d}{T}}$

Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency of a vibrating string is determined by the relationship f = 1/(2L) ?(T/?), where L is the length of the string, T is the tension applied, and ? is the linear mass density of the string. This equation arises from the study of standing waves on a string and is a fundamental concept in wave mechanics, reflecting how the physical properties of the string and the applied tension influence its oscillatory motion.

Linear Density and Cross-Sectional Area

Linear density (?) is defined as the mass per unit length of a string and can be determined by the product of the material's density (d) and its cross-sectional area (A). For a wire with a circular cross-section, the area is given by A = ?r², where r is the radius of the wire. Understanding this relation is crucial when substituting ? in terms of density and geometry into the fundamental frequency formula, linking the material properties to the vibrational characteristics.

Tension and Wave Propagation

Tension in the string plays a critical role in the propagation of waves along the string. An increase in tension leads to a higher wave speed, which in turn raises the frequency of the standing wave patterns that can be set up on the string. This dependency of frequency on tension is a core concept in studying the dynamics of vibrating strings and is instrumental in applications like musical instruments and engineering devices.

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