# Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer

## What is Sonometer?

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.

### Materials Required

The following materials are necessary for the experiment.

• A Sonometer

• 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:

$\nu = \frac{1}{lD}\sqrt{\frac{T}{\pi \rho}}$

where $\nu$ is the frequency of the string, l is the length of the string, D is the diameter, $\rho$ is the material density and T is the tension.

The above equation clearly shows that $\nu$l = 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 $\nu$ and 1/l will also be a straight line.

### Procedure

1. Place the sonometer on the table and add the weight of 4kg.

2. It is essential to use a frictionless pulley for this experiment. Moreover, the maximum weight carried by the hanger must be suitable for it.

3. Move the wooden bridges accordingly to get the maximum length of the wire.

4. Now, select 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.

5. Plug the sonometer wire to generate vibration in it. Compare the sound produced by the vibration of plugged wire and tuning fork.

6. Adjust the length of the sonometer wire using wooden bridges. Adjust until the sounds of vibrations from both the sources look similar.

7. Now, place the paper rider in the middle of the wire of the sonometer.

8. 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.

9. 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.

 Sl No Load M (kg) Tension Mg T (N) Resonant Length of wire l2 (cm2) l2/T (cm2N-1) Length Increasing l1 (cm) Length decreasing l2 (cm) Mean =(l1+ l2) /2 l cm 1 4.0 2 3.5 3 3.0 4 2.5 5 2.0 6 1.5 7 1.0

After observation and calculation, plot a graph between T and l2, taking T along x-axis and l2 along the y-axis.

The above image shows the graph between T and l2, which is a straight line.

### 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.

1. What is The Law Of Vibrating String?

The fundamental frequency of vibration of a stretched wire or string is 𝝂= (1/2l) √(T/m). This expression helps us to derive three laws of vibrating string.

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?

Many people think that both phenomena belonging to the sound are the same. However, they are different from each other with respect to the tone quality. In simple words, the pitch of any sound is the responsiveness of the ears to the sound wave. Every individual has a different voice quality due to different pitch or tone. The loudness of the sound depends on the energy of the sound wave. The loudness of the sound depends on the amplitude while the pitch depends on its frequency.