Question

Fundamental frequency of a sonometer wire is \(\mathrm{n}\). If the length and diameter of the wire are doubled keeping the tension same, the new fundamental frequency is...... (A) \((2 \mathrm{n} / \sqrt{2})\) (B) \(\\{\mathrm{n} /(2 \sqrt{2})\) (C) \(\sqrt{2 n}\) (D) \((\mathrm{n} / 4)\)

Short answer

The new fundamental frequency is \(\dfrac{n}{4}\).

Step-by-step solution

Recall the fundamental frequency formula

The formula for the fundamental frequency of a vibrating string is given by: \(f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}\) Here, \(f\) is the fundamental frequency, \(L\) is the length of the string, \(T\) is the tension, and \(\mu\) is the linear mass density.

Determine the linear mass density of the original wire

Since the problem only provides us with the diameter change, we will first find the relationship between linear mass density and the diameter. Linear mass density (\(\mu\)) can be calculated as: \(\mu = \dfrac{m}{L}\) Where \(m\) is the mass of the wire, \(L\) is the length of the wire, and the mass of a cylinder (the wire) is given by: \(m = \pi r^2 h \rho\) Here \(r\) is the radius (half of the diameter), \(h\) is the height (which is equal to length in this case), and \(\rho\) is the density of the material. When we substitute the mass formula into the linear mass density formula, we will have: \(\mu = \dfrac{\pi r^2 L \rho}{L}\) This will simplify to: \(\mu = \pi r^2 \rho\)

Write the new linear mass density and length

Since the length and diameter are doubled, let the new length be \(2L\), and the new diameter be \(2d\), where \(d\) is the original diameter. The new radius is thus equal to \(r_{new} = 2r\), thus the new linear mass density (\(\mu_{new}\)) will be: \(\mu_{new} = \pi (2r)^2 \rho = 4\pi r^2 \rho\)

Find the new fundamental frequency

Substitute the new length and linear mass density into the fundamental frequency formula: \(f_{new} = \dfrac{1}{(2)(2L)} \sqrt{\dfrac{T}{4\pi r^2 \rho}}\) Factor out the ratio of the new frequency to the original frequency: \(\dfrac{f_{new}}{n} = \dfrac{1}{(2)(2L)} \sqrt{\dfrac{T}{(4\pi r^2 \rho)(2L)}}\) The original fundamental frequency can be written as: \(n = \dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r^2 \rho}}\) We can now write the ratio of the new frequency to the old frequency as: \(\dfrac{f_{new}}{n} = \dfrac{1}{(2)(2L)} \sqrt{\dfrac{T}{(4\pi r^2 \rho)(2L)}} \times \dfrac{\sqrt{\dfrac{T}{\pi r^2 \rho}}}{\dfrac{1}{2L}}\) After simplification, we find: \(\dfrac{f_{new}}{n} = \dfrac{1}{4}\) Finally, we can find the new fundamental frequency: \(f_{new} = \dfrac{n}{4}\) The correct answer is (D) \(\dfrac{n}{4}\).

Key concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system naturally vibrates. It is the simplest mode of vibration, also known as the first harmonic or natural frequency.
For stringed musical instruments or devices like a sonometer, the fundamental frequency determines the pitch of the sound that the wire produces when it vibrates. In a sonometer wire, the tension, length, and the material properties of the wire affect the fundamental frequency.
The fundamental frequency of a vibrating string can be calculated using the formula:

• \(f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}\)

where \(f\) stands for fundamental frequency, \(L\) is the length of the string, \(T\) is the tension force, and \(\mu\) is the linear mass density.Understanding this basic concept helps in manipulating the properties of the wire, such as its length and tension, to achieve the desired frequency.

Vibrating String Formula

The vibrating string formula is critical in understanding how strings on instruments like the sonometer produce sound. This formula connects the physical properties of the string with the frequency of the sound it creates:

• \(f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}\)

This formula highlights several key factors:

• Length \(L\): Longer strings typically produce a lower frequency.
• Tension \(T\): Higher tension results in a higher frequency.
• Linear Mass Density \(\mu\): This is a measure of mass per unit length. Heavier strings produce lower frequencies.

By changing these variables, one can control the sound waves the string produces. The formula is fundamental not only for scientific purposes but also for practical applications like music tuning.

Linear Mass Density

Linear mass density, represented by \(\mu\), is a vital concept in understanding vibrating systems such as sonometer wires. It is defined as the mass per unit length of the wire:

• \(\mu = \dfrac{m}{L}\)

For a cylindrical wire, such as our problem's wire, the mass \(m\) can be expressed as:

• \(m = \pi r^2 h \rho\)

Here, \(\pi r^2\) represents the cross-sectional area of the wire, \(h\) is the height or the length of the wire, and \(\rho\) is the density of the wire material. Simplifying, the formula for linear mass density becomes:

• \(\mu = \pi r^2 \rho\)

Thus, any changes in radius or density will affect the linear mass density, subsequently changing the wire's vibrational characteristics. For example, doubling the diameter increases the cross-sectional area fourfold, altering the frequency of vibration by increasing \(\mu\).

Sonometer Wire

A sonometer is a device used to study the transverse vibrations of strings. It consists of a wire stretched over a resonating box or board, with weights applied to one end to adjust tension. The sonometer wire, therefore, is crucial in understanding how length, tension, and mass density interact to affect vibration.
In practical terms, a sonometer allows for the precise study of wave properties in a controlled environment. Key components, such as the length of the wire and the force applied, can be easily adjusted to observe changes in vibration frequency. By experimenting with a sonometer, students can gain a hands-on understanding of concepts like fundamental frequency and linear mass density as they see the direct impacts of changing these variables.
Moreover, sonometers are aligned with the vibrating string formula, \(f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}\), allowing accurate predictions of frequency changes based on modifications in wire properties. This makes the sonometer a valuable educational tool in exploring the science of sound and wave mechanics.