How Does Changing Wire Length Affect Tension When Frequency Stays Constant?
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 Wire Length and Tension at Constant Frequency
1. What is a sonometer and what is its main purpose in physics experiments?
A sonometer is a laboratory instrument used to study the properties of vibrating strings. Its main purpose is to demonstrate and verify the relationship between the frequency of the sound produced by a string, its length, its tension, and its mass per unit length. It is commonly used to verify the laws of transverse vibrations in strings.
2. What are the three laws of vibrating strings that the sonometer helps to verify?
The sonometer is used to verify three fundamental laws of vibrating strings:
- Law of Length: The fundamental frequency of a string is inversely proportional to its length (f ∝ 1/l), provided the tension and mass per unit length are constant.
- Law of Tension: The fundamental frequency is directly proportional to the square root of the tension in the string (f ∝ √T), provided the length and mass per unit length are constant.
- Law of Mass: The fundamental frequency is inversely proportional to the square root of the mass per unit length (f ∝ 1/√m), provided the length and tension are constant.
3. In a sonometer experiment, what are nodes and antinodes?
In the standing wave formed on a sonometer wire, nodes are the points that remain stationary and have zero amplitude (no vibration). In contrast, antinodes are the points located midway between the nodes where the amplitude of vibration is maximum. The distance between two consecutive nodes is always half a wavelength.
4. How does increasing the tension in a sonometer wire affect the pitch of the sound produced?
Increasing the tension in a sonometer wire makes the sound's pitch higher. This is because pitch is directly related to frequency. According to the law of tension, the frequency of vibration is proportional to the square root of the tension (f ∝ √T). Therefore, a tighter string vibrates faster, producing a higher frequency and, consequently, a higher pitch.
5. Why is a paper rider used in the sonometer experiment? What does it indicate?
A small, lightweight piece of paper, called a paper rider, is placed on the sonometer wire to find the exact point of resonance. When the frequency of the tuning fork matches the natural frequency of the wire segment, the wire vibrates with maximum amplitude. This intense vibration throws the paper rider off the wire. This event indicates that resonance has been successfully achieved.
6. If the length of a sonometer wire is halved and the tension is doubled, what happens to the fundamental frequency?
The fundamental frequency will increase significantly. The frequency (f) is proportional to the square root of tension (√T) and inversely proportional to length (l). Let's see the combined effect:
- Halving the length (l/2): This causes the frequency to double.
- Doubling the tension (2T): This increases the frequency by a factor of √2.
Therefore, the new frequency will be 2 multiplied by √2, making it 2√2 times higher than the original frequency.
7. What is the difference between harmonics and overtones in the context of a vibrating string?
While often used interchangeably, harmonics and overtones have distinct meanings for a vibrating string:
- Harmonics are all the possible frequencies a string can produce, which are integer multiples of its lowest frequency (the fundamental). The fundamental frequency itself is called the first harmonic.
- Overtones are all the frequencies produced by the string that are higher than the fundamental frequency.
For a string fixed at both ends, the first overtone is the second harmonic, the second overtone is the third harmonic, and so on.
