A sonometer wire supports a 4 kg load and vibrates in fundamental mode with a tuning fork of fr

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A sonometer wire supports a 4 kg load and vibrates in...

A sonometer wire supports a $4 \mathrm{~kg}$ load and vibrates in fundamental mode with a tuning fork of frequency $416 \mathrm{~Hz}$. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to $\ldots \ldots \ldots$ (A) $1 \mathrm{~kg}$ (B) $2 \mathrm{~kg}$ (C) $8 \mathrm{~kg}$ (D) $16 \mathrm{~kg}$

A sonometer wire supports a $4 \mathrm{~kg}$ load and vibrates in fundamental mode with a tuning fork of frequency $416 \mathrm{~Hz}$. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to $\ldots \ldots \ldots$
(A) $1 \mathrm{~kg}$
(B) $2 \mathrm{~kg}$
(C) $8 \mathrm{~kg}$
(D) $16 \mathrm{~kg}$

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Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency is the lowest frequency at which a string vibrates and is defined by the relationship f = 1/(2L) ?(T/?), where L is the length of the vibrating string, T is the tension, and ? is the linear mass density. This equation illustrates how the properties of the string determine its vibrational characteristics.

Relationship Between Tension, Length, and Frequency

Since the fundamental frequency depends on tension and length, any change in the length of the string requires an adjustment in tension to keep the frequency constant. Specifically, if the length is doubled, the tension must be reduced by a factor of four, because the frequency is inversely proportional to the length and proportional to the square root of the tension.

Tension Calculation Through Load

In practical applications, the tension in a string is generated by the load attached to it (with T = mg). Adjusting the load directly changes the tension. Understanding this connection is essential in experiments with vibrating strings, because maintaining a constant frequency when changing the string length necessitates recalculating the appropriate load to achieve the required tension.

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