Sonometer – How to Verify the Laws of Vibration in a Fixed String

A sonometer is a device used to verify the laws of vibration in a fixed string. It consists of a one-meter long hollow box with a consistent metal wire fastened inside.

The frequency of the transverse standing wave in the wire is directly proportional to its tension and inversely proportional to its length and mass per unit length (linear density). To prove this, the wire with different linear densities l and m is kept under constant tension T.

What is a Sonometer?

A sonometer is an instrument used to demonstrate the relationship between the frequency of a plucked string, and the length, tension and mass per unit length of the string. These relationships are usually called Mersenne’s Law, after Marin Mersenne (1588-1648) who investigated and codified them.

A simple sonometer consists of a hollow one-meter-long box with a uniform metallic thin string attached to it. One end of the string is attached to a hook and the other is connected to a weight hanger through a pulley. Iron weights can be hung from the holder to vary the wire tension. Two adjustable wooden knives are put over the board, and their positions can be adjusted to change the vibrating length of the string.

The string is then gently plucked to produce a transverse standing wave sound in the board. The sound is heard in the knife edges (nodes) and antinodes. When the string is elongated, a new set of nodes and anti-nodes are formed. This is because the resonating frequency and amplitude of a stretched string varies inversely with its length and tension.

This experiment is easy to perform, but it is important to use a high-quality string. The vibrations of a low-quality string can damage the board. It is also important to use a good quality tuning fork and a high-quality pulley.

To set up a sonometer, first put a piece of masking tape on the sonometer board under String B. Make sure that the tape does not touch the sonometer pegs. Next, place a tuning fork over the end of the sonometer string. A musical note C at 256 Hz is recommended.

When the tuning fork is struck, the board will resonate. The frequency of the resonance is recorded on a note sheet. It is then used to calculate the linear density of the sonometer wire.

The sonometer is a modified version of the monochord, which was invented by Pythagoras in the 7th century BC. A French instrument maker, Albert Marloye, modified the monochord into a differential sonometer in the mid 1800’s. This device is still widely used in physics labs today.

Uses of a Sonometer

A sonometer is a device that allows you to verify the laws of vibration in a fixed string. It shows the connection between the frequency of the sound produced by a plucked string and its tension, length and mass per unit length. This is achieved by producing a transverse standing wave in the string which produces sound. It is also used to determine the frequency of a tuning fork and the unknown hanging masses.

A typical sonometer consists of a one-meter long hollow box with consistent metallic rope fastened to it with the help of a pulley. One end of the string is attached to a hook while the other end is connected to a weight hanger through the pulley. The free end of the string is equipped with weights to enhance its tension. Two adjustable knives are placed on the board and their positions can be modified to change the vibrating length of the stretched string.

The first law of the vibrating string states that the frequency of the string varies inversely with its resonating length. It is further proven by the fact that the second law of the vibrating string states that the fundamental frequency of a string is inversely proportional to its linear density if the length and the tension are constant. This is confirmed by the fact that a given string of equal length can be made to vibrate in resonance with different tuning forks having different frequencies by altering its tension.

It is important to note that when the string is plucked, it starts vibrating at many of its natural resonance frequencies at the same time. Which ones are chosen depends on the initial displacement of the string. This displacement is determined by the shape of the initial vibration wave pulse.

Another use of a sonometer is to confirm the third and fourth laws of a vibrating string. The third law states that a plucked string will produce a standing wave in the form of nodes and anti-nodes, which are the points at which the incoming sound waves meet. The fourth law explains that the amplitude of a standing wave in a stretched string is directly proportional to its frequency.

Laws of Vibration

The laws of vibration can be verified using a sonometer to measure the vibrations in a fixed string. These laws state that the frequency of a vibrating body depends on its linear density and the frequency of the source. These frequencies are usually in integer multiples of the natural frequency of the fixed string. The laws of vibration also state that the vibrating length and mass per unit length of the string must be constant.

A transverse standing wave is formed when the string is plucked. This wave is reflected off the end of the string, and the displacement at that location is zero. The waves are then reflected back and forth between the ends of the string, creating nodes and anti-nodes in the wave. The vibrations of the string can be measured by observing the movement of the nodes and anti-nodes.

To verify the laws of vibration of a fixed string, the sonometer is used to measure the resonating length and mass per unit length of the wire. The sonometer is made up of a one meter long hollow box. An experimental wire AB is set into resonance with the help of an auxiliary wire PQ with known tension. The resonating length of the experimental wire is adjusted until resonance occurs with the frequency of the tuning fork. The resonating length of the auxiliary wire is noted as L1. The experimental wire is then plucked at a frequency of 100 Hz, and the resulting sound is recorded on a recorder.

This reveals that the resonating length of the experiment is constant, and the frequency of the experimental wire is directly proportional to the square root of its linear density. This verifies the first law of vibrations in a fixed string.

To verify the second law of vibrations in a fixed string, the sonometer is equipped with a set of tuning forks that have different frequencies. The resonating lengths of each tuning fork are determined by measuring their frequencies and their respective resonances. The resonating lengths of the experiment and each tuning fork are then calculated. It is found that the resonating lengths of the experiment are identical to the corresponding resonant lengths of the tuning fork, within experimental error. This demonstrates that the law of linear density is valid.

Vibrations in a Fixed String

The strings of musical instruments are fixed at both ends and, therefore, they can only vibrate at certain modes. The frequencies of these modes are harmonically related, with each successive mode having a frequency twice the previous one. This series is called the harmonics. The first four harmonics are given by the formula: f = (v/l)/2, i.e. f(v/l)/2 = 1 – (v/l)/2.

A string fixed at both ends can also have a standing wave pattern which occurs at a frequency of twice the fundamental. This results in nodes at each end and antinodes in the middle of the string. This is a common wave shape in many musical instruments.

This type of vibration is also called forced oscillations. It is important to note that the amplitude of this type of vibration will be proportional to the square root of the tension of the string. Therefore, it is important to keep the amplitude of the string constant when doing this experiment.

Another way to verify the laws of vibration in a fixed string is to use a sonometer. This instrument will measure the frequency of the vibrations and will provide you with the corresponding amplitude. This will allow you to determine whether or not the vibrations of the fixed string are proportional to the amplitude of the tuning fork.

For this experiment, you will need a tuning fork with a known frequency and two wires with different masses per unit lengths m1 and m2. The first wire is subjected to a suitable amount of tension and made to vibrate in unison with the tuning fork. The resonating length of the first wire is noted as (l1). The second wire is then made to vibrate with the same tuning fork and the resonating length of the second wire is noted as (l2). The resonant frequencies of the two wires are then compared and, within experimental errors, it should be found that n = n1l2 and n = n2l1 if the tension of each wire and its mass per unit length remain constant.

If you are unable to get a sonometer, you can try this experiment using a long spring such as a slinky or several metres of flexible rubber hose. The key is to find a spring which is stretched a little so that it produces a kink in the middle. Once you have done this, the slinky or flexible hose can be set into motion by a quick pull.