Sonometer - Theory, Calculations and Precautions to be Taken
A sonometer is a device used in illustrating the relationship between tension in a string, length and frequency of the sound. It comprises a hollow rectangular wooden box which has a length of more than one meter. One end of this device consists of the hook while the other end comprises the pulley. The first end of the string present in the sonometer is attached to the hook while the other end passes over the pulley. At the free end of the string or wire, A weight hanger is present where you can attach weights. Two adjustable bridges are also present above the board that allow the user to adjust the length of the wire or string. In this article, we will study the relationship between tension for constant frequency in a given wire along with the length using a sonometer.
What is the Law of Tension?
In Sonometer, the law of tension refers to the direct proportionality of the frequency of sound vibrations in a string, to that of the square root of the tension produced in the vibrating string and mass per unit length are constant. The law can be verified by keeping one of the variables constant consecutively while one determines the relationship between the other two variables. The law is experimentally verified in the experiment stated above.
Principle of Sonometer
Sonometer (‘sonos’ meaning sound, ‘metric’ meaning measurement) is a diagnostic device that is used to measure the frequency of a sound wave. It is variously used in sound vibration systems, such as in this experiment, to study the relationship between frequency, tension, density and length of a stretched string. In sound mechanics, resonance refers to the maximum increase in amplitude in response to the coherence of sound waves oscillating in natural frequency, such that the total amplitude is greater than the sum of its parts. In other words, resonance occurs when there is ‘positive interference’ of sound waves.
Sonometer also has several practical uses. In the medical field, it is extensively used to measure bone mineral density by the use of ultrasound energy.
Aim of the Experiment
To experimentally study the Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer.
Materials Required
The following materials are necessary for the experiment.
A Sonometer
Rubber pad
Screw gauge
½ kg hanger
Meter- scale
A set of eight tuning forks
Seven ½ kg slotted weights
Paper rider
Theory
If the stretched wire (string) and tuning fork vibrate in the same resonance, then their frequency will be the same. The frequency of the string can be given by using the law of vibrating string as follows:
ν = \[\frac{1}{lD}\]\[\sqrt{\frac{T}{\pi \rho}}\]
where ν is the frequency of the string, l is the length of the string, D is the diameter, ρ is the material density and T is the tension.
The above equation clearly shows that νl = constant. It indicates that T ∝ l², which means that the graph between T and l² will be a straight line. The graph between v and 1/l will be a hyperbola and the graph between ν and 1/l will also be a straight line.
Procedure
Place the sonometer on the table and add the weight of 4kg.
It is essential to use a frictionless pulley for this experiment. Moreover, the maximum weight carried by the hanger must be suitable for it.
Move the wooden bridges accordingly to get the maximum length of the wire.
Now, select a 256 Hz fork from the set of tuning forks and make it vibrate by striking it against the rubber pad. Bring the fork near to your ear after hitting.
Plug the sonometer wire to generate vibration in it. Compare the sound produced by the vibration of plugged wire and tuning fork.
Adjust the length of the sonometer wire using wooden bridges. Adjust until the sounds of vibrations from both the sources look similar.
Now, place the paper rider in the middle of the wire of the sonometer.
Hit the tuning fork using the rubber pad to produce vibrations and place its striking side on the first end of the sonometer where the string is attached. Slowly adjust the position of the second wooden board until the paper rider falls off from it and measure the length of the string.
Repeat the same process with reduced weights of 3.5kg, 3 kg, 2.5 kg until you will reach 1kg. Record the measurements in the tabular form with every load.
Observations and Calculations
Frequency of the tuning fork in every case= 256 Hz. Fill the sonometer experiment readings in the below table. After observation and calculation, plot a graph between T and l2, taking T along the x-axis and l2 along the y-axis. The above image shows the graph between T and l2, which is a straight line.
Calculations:
First find the mean length ‘l’, and note it down in the observation table. Mean length can be calculated as : sum of l1(increased length) and l2 (decreased length) divided by 2.
Then find the square value of ‘l’ ( l2) and note it down in the given column.
Then after, divide the value obtained in step 2 by the ’T’ (tension) and note in the given column.
Finally plot a graph taking ‘T’ along the X-axis and l2 along the Y-axis. The straight line obtained in the graph shows the relation between the experimental variables.
Result
From the above graph, it is clear that l2/T = constant which concludes that T ∝ l2. It verifies the law of vibrating string and sonometer formula.
Precautions
Pulleys used during the experiment must be frictionless.
The wires used for the experiment must be kinkless and of uniform cross-section.
The maximum weight used in the experiment must not go beyond the elastic limit.
The soft rubber pad is ideal for striking the tuning forks for causing vibrations.
Make sure to touch the lower end of the tuning fork gently with a sonometer board to transfer the waves.
Don't forget to include the hanger weight in the load during the experiment.
Remove the load carefully after completing the experiment.
Sources of Error
Following can be the sources of error for this experiment:
Make sure to use a metal wire of uniform length, thickness and cross-sectional area.
Make sure the wire is rigid in form.
The weights may be unknown or inaccurate.
Pulley may not be frictionless.
The wire used has kinks.
A soft rubber pad is not used
FAQs on Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer
1. What is The Law Of Vibrating String?
The fundamental frequency of vibration of a stretched wire or string is 𝝂= \[(\frac {1}{2l})\] \[\sqrt {\frac {T}{m}}\] . This expression helps us to derive three laws of vibrating strings.
Law of Length: The fundamental frequency is inversely proportional to the vibrating length if mass per unit length and tension remains constant
Law of Tension: If we keep length and mass per unit length constant in a vibrating string, then its frequency will be directly proportional to the square root of applied tension.
Law of Mass: if we keep length and tension constant in a vibrating string, then the frequency will be inversely proportional to its square root of mass per unit length.
2. What is The Difference Between The Loudness And Pitch Of A Sound?
Loudness and Pitch are both different attributes of sound waves. The loudness of a sound wave is determined by the maximum amplitude (particle displacement) across a medium. If the amplitude is large, the sound is said to be loud and vice versa. Loudness is expressed in decibel (dB). All higher animals can hear/ detect sounds of different frequencies. Human beings have a range beyond which sound is indetectable or considered to be noise. Sound beyond 80 dB is considered a nuisance to the human ear.
Pitch, on the other hand, is determined by the frequency of a sound wave. Biologically, women have higher pitch sound compared to men due to sexual dimorphism. (To know more about the laws governing the sound waves, refer to the online preparatory study material for Physics).
3. What is the “superposition” of sound waves? State its principle.
The phenomenon of coherence of two or more than two sound waves to produce a new wave is known as superposition of sound waves. A positive coherence/superposition produces the effect of resonance. It is based on the displacement of particles, that equals the vector summation of individually displaced particles in space giving rise to “superposition”. When these super imposing waves travel with the same velocity, frequency and amplitude, but in opposing directions, they give rise to stationary waves.
Example: When vibrations travel along a string held firmly at both the ends.
4. What are Nodes? How is the distance between two consecutive nodes measured?
In sound mechanics, nodes refer to the points in stationary waves that show zero net displacement. These are regions of maximum strain as the waves travel across the medium. Nodes are represented by the letter ‘N’. And the distance between any two consecutive nodes in such stationary waves is equal to half of the wavelength of the waves. The opposite of a ‘Node’ is called the ‘Antinode’ and it denotes the region of minimum strain and maximum net displacement.
5. Define Harmonics.
Harmonics is an important concept in the field of sound mechanics as it is extensively used in instruments that produce sound and especially those that involve creation of rhythm or music. In mechanical terms it refers to frequencies that are integral multiples of a fundamental tone, where the fundamental itself is the first harmonic (as it is a multiple of itself). A set of harmonics creates a ‘Harmonic series’ or ‘Overtone’. The series then follows from fundamental frequency or first harmonic to second harmonic (which is twice the fundamental frequency), third harmonic (thrice the fundamental frequency), and so on. Harmonics determine the quality of sound.
