Relationship Between Frequency and Length of Wire under Constant Tension Using Sonometer

Relationship between the frequency and the length of the wire under constant tension can be experimentally established using a sonometer. If a stretched wire or string starts vibrating at the resonance with a tuning fork of frequency ‘ν’, then the string also will have the same frequency of ‘ν’. Sonometer is a device used for demonstrating the relationship between tension in a string, length, mass per unit length of the string and frequency of the sound that is produced by the string when it is plucked.

Also Read: Speed of Sound

Key Takeaways- Sonometer, Frequency, Resonance, Tension, Interference

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What Is Sonometer?

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Sonometer is a device used for demonstrating the relationship between tension in a string, length, mass per unit length of the string and frequency of the sound that is produced by the string when it is plucked. It is an ancient musical instrument but also used as a scientific laboratory instrument. 

Sonometer

It comprises a hollow wooden box having a length of more than one meter. One end of the device has a hook while the other end comprises the pulley. Here, the first end of the string present in the sonometer is attached to the string and the other end passes over the pulley. A weight hanger is present where the weights can be attached. There are two adjustable bridges present above the board that allow the user to adjust the length of the string.

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Principle of Sonometer

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Sonometer is a device that is used to measure the frequency of the sound wave. It is a widely used device in sound mechanics. The fundamental principle of the device is ‘resonance’. Resonance refers to the maximum increase in amplitude in response to the coherence of sound waves oscillating in natural frequency in such a way that total amplitude is always greater than the sum of its parts. It occurs when there is a positive interference of sound waves.

Resonance

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Law Of Tension

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In Physics, tension is described as a pulling force transmitted by means of a string or a similar object. It can also be described as the action-reaction pair of forces working at the end of each element.

Tension

In Sonometer, the law of Tension refers to the fundamental frequency of sound vibrations of a string that is directly proportional to the square root of tension if the vibrating length and mass per unit length remain constant. The law can be easily verified by keeping one of the variables constant and using the other one to determine the relationship between the other two variables.

ν ∝ t (When l and m are constant)

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Relationship Between Frequency and Length of Wire under Constant Tension- Experiment

Aim

To experimentally examine the Relationship Between Frequency and Length of Wire under Constant Tension Using Sonometer.

Required Instruments

The following things are required for the experiment

• A Sonometer
• A Rubber pad
• ½ kg hanger
• Screw Gauge
• Meter scale
• A set of eight tuning forks
• Seven ½ kg slotted weights
• Paper rider
• Paper, pencil and graph paper

Sonometer Experiment Material Needed

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Theory

If the stretched wire or string starts vibrating at the resonance with a tuning fork of frequency ‘v’, then the string also will have the same frequency of ‘v’.

Let us assume that the string has a length of ‘l’ and diameter of ‘D’, its density is ‘ρ’ and the tension is ‘T’, then

v = 1ldTπρ…..(1)

Relation between frequency and length (l)

From the above equation, we get, 

v∝ 1l

or 

vl = constant

Thus, a graph between v and 1/l will be a straight line and a graph between v and l will be hyperbolic.

Relation between length and tension (T)

From equation 1 we get,

Tl = constant

Or

T∝ l

Or, 

T∝ l2

Here, a graph between T and l2 will be a straight line.

Experimental Setup

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Procedure

• The sonometer is placed on the table and a weight of 4 kg is added.
• Here, it is absolutely crucial to use a frictionless pulley. The maximum weight carried by the hanger should be suitable.
• Now, the wooden bridges are moved accordingly to get the maximum length of the wire.
• 256 Hz fork from the set of tuning forks is selected and it is vibrated by striking it against the rubber pad. It is brought near to the ear after hitting.
• The sonometer plug is wired to generate vibration in it. The sound produced by the vibration of the plugged wire and tuning fork is compared.
• The length of the sonometer wire is adjusted using the wooden bridges. It is adjusted until the sounds of the vibrations from both sources look similar.
• Now, the paper rider is placed in the middle of the wire of the sonometer.
• Now, the tuning fork is hit using the rubber pad to produce vibrations and it is placed with its striking side on the first end of the sonometer where the string is attached. 
• The position of the second wooden board is adjusted until the paper rider falls off it and the length of the string is measured.
• The whole process is repeated with reduced weights of 3.5kg, 3 kg, 2.5kg and so on until 1kg is reached. 
• The measurement from every load is tabulated.

Observations and Calculations

Constant tension on the wire, T=____kg.

Table of the frequency and length

Calculations

1. Find the mean length (l)
2. Determine 1/l
3. Plot the graph between ν and l putting ν along the X axis and l along the Y axis.
4. Plot the graph between ν and 1/l by putting ν along the X axis and l along the Y axis.

Result

From the above graph, it is clear that the graph of νl= constant and v∝1l

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Precautions

The following precautions should be taken in the experiment

• Pulleys used in the experiment should be frictionless.
• The wires should be kinkless and of uniform diameter.
• The maximum weight used should not go beyond the elastic limit.
• The loads should be carefully removed after completing each set of experiments.

Read Also: Photoelectric Effect

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Things to Remember

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• Sonometer is a device that is used to measure the frequency of the sound wave. Here, the fundamental principle of the device is ‘resonance’.
• Tension can be described as a pulling force transmitted by means of a string or a similar object.
• The instruments needed for the experiment are a Sonometer, a Rubber pad, ½ kg hanger, Screw Gauge, meter scale, a set of eight tuning forks, Seven ½ kg slotted weights, paper rider, Paper, pencil and graph paper.
• A graph between v and 1/l will be a straight line and a graph between v and l will be hyperbolic and the graph between T and l2 will be a straight line.
• In this experiment, Pulleys used in the experiment should be frictionless and the wires used should be of uniform diameter.

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