Two wires made up of same material are of equal lengths but...

Two wires made up of same material are of equal lengths but their radii are in the ratio $1: 2$. On stretching each of these two strings by the same tension, the ratio between their fundamental frequency is $\ldots \ldots \ldots .$
(A) $1: 2$
(B) $2: 1$
(C) $1: 4$
(D) $4: 1$

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Key Concepts

Effect of Cross-Sectional Area on Linear Density

Since the cross-sectional area of a wire is given by A = ?r², a change in the radius leads to a quadratic change in the area and hence the linear density. In problems involving wires or strings of the same material and length, a wire with a larger radius will have a higher linear density, resulting in a lower fundamental frequency under the same tension compared to a wire with a smaller radius.

Fundamental Frequency of a Vibrating String

The fundamental frequency of a stretched string is given by the formula f = (1/2L)?(T/?), where T is the tension in the string, L is its length, and ? (mu) is the linear mass density. This relationship shows that the frequency increases with higher tension and decreases with greater mass per unit length.

Linear Density

Linear density, denoted by ?, is the mass per unit length of the string. For a wire made of a uniform material, the linear density is directly proportional to its cross-sectional area. Therefore, if the radius of the wire is changed, the linear density changes according to the square of the radius, affecting the vibration frequency.

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