# How to measure the AC source frequency using sonometer and electromagnet

In this project, I explain how to measure the AC mains frequency using sonometer and electromagnet in the lab. Basically the frequency f of a stretched string depends upon the length of the string l, the tension T, and the mass per unit length m. Defined by relation f=1/2l v(T/m). The wire is made to vibrate by passing AC current of unknown frequency and when frequency of stretched string is in resonance with AC source frequency then the frequency of AC mains n is equal to the frequency f of wire.

## What is sonometer?

A sonometer is a hollow wodden box used in lab to study the transverse vibrations of string. It is also called the monochord because it often has only one string. On the wooden rectangular box are two fixed bridges, near the ends, and at one end is a pulley. A string, often a steel wire is fastened at one end run over the bridges and the pulley, and attached to a weight holder hanging below the pulley. Weights can be added to the holder to produce tension in the wire, and a third movable bridge can be placed under it to change the length of the vibrating section of the string.
A sonometer demonstrates the relationship between the frequency produced by a plucked string,tension, length and mass per unit length of the string.
For small amplitude vibration, the frequency is proportional to:
a) The square root of the tension of the string,
b) The reciprocal of the square root of the linear density of the string,
c) The reciprocal of the length of the string.

f=1/2l v(T/m)

## What is electromagnet?

A wire with an electric current passing through it generates a magnetic field around it, this is a simple electromagnet. The strength of magnetic field generated is proportional to the amount of current.
In order to concentrate the magnetic field generated by a wire, it is commonly wound into a coil, where many turns of wire sit side by side. The magnetic field of all the turns of wire passes through the centres of the coil, creating a strong magnetic field there. A coil forming the shape of a straight tube (a helix) is called a solenoid, much stronger magnetic fields can be produced if a core of ferromagnetic material, such as soft iron is placed inside the coil. The ferromagnetic core magnifies the magnetic field to thousands of times the strength of the field of the coil alone, due to the high magnetic permeability(µ) of the ferromagnetic material. This is called a ferromagnetic-core or iron-core electromagnet.

The direction of the magnetic field through a coil of wire can be found from a form of the right-hand rule. The main advantage of an electromagnet over a permanent magnet is that the magnetic field can be rapidly manipulated over a wide range by controlling the amount of electric current. However, a continuous supply of electrical energy is required to maintain the field.

## Theory of a sonometer

If a sonometer wire of length l stretched between two knife-edges under a tension T and mass per unit length m is once plucked and then released it executes transverse vibrations and emits a note of fundamental frequency n Hz given by

n=1/2l v(T/m)

To find the frequency of A.C. main using an electromagnet and a sonometer, the A.C. is passed through the primary of a step-down transformer (220-230 to 4-6 volts). The two ends of the secondary coil of the step-down transformer are connected to the two ends of the sonometer wire, and D.C. is passed through the windings of the electromagnet which consists of a coil of insulated copper wire wound over a soft iron core provided with an inulated handle.

As the D.C. passes from windings of electromagnet it gets magnetised with its faces as North and South Pole. The electromagnet is kept close to and vertically above the sonometer steel wire. As the A.C. from the secondary of the step-down transformer passes through the sonometer wire it starts vibrating.

The position of the knife edges is so adjusted that the sonometer steel wire vibrates in resonance with the A.C. supply (In India A.C. supply is 50 Hz).
Hence,
n=f=1/2l v(T/m)

## Procedure for measuring the AC frequency using a sonometer

1. Set the sonometer as explained.
2. Arrange the electromagnet in a clamp stand and hold it 2-3 mm vertically above the center of the steel wire of the sonometer. Connect the sonometer to secondary of the step-down transformer and connect the D.C. supply to electromagnet, switch on the D.C. and test the magnetization of the electro-magnet with the help of iron needle.
3.Cut a V-shaped light paper rider about one cm long and 2 mm wide. Bring the two knife edges close to each other and place the rider on the wire in between the knife edges. See that the pole of the electromagnet is just above the center of the sonometer wire.
Now gradually increase the distance the two knife edges till the rider begin to flutter. At this stage, make the adjustment very carefully till the rider flies off at a certain fixed distance between the two knife edges. The wire is now in resonance with the frequency of the A.C. mains supply. Measure the length of the wire between the knife edges with a meter rod.
4. Increase the distance between the two knife edges by a few centimeter. Repeat the above process by decreasing slowly the distance between the two knife edges till the rider again flies off. Measure the length of the wire between the two knife edges again. The mean of the two length is the true resonant length.
5. Weigh the weights suspended including the hanger with a trip-scale balance.
6.Increase the weight by half a kilogram and repeat the observation to find the length of the wire vibrating in resonance with A.C. mains supply. Take such four observations by changing the load by 1/2 kg each time.
7. Remove the wire to find the mass per unit length of the wire used. Measure the length of the wire and weight accurately by a sensitive balance.

## Precautions

1. The wire should be of a uniform area of cross-section, free from kinks and should be taut.
2. The observation should start with minimum distance between the two wedges.
3. The resonance position should obtained by first lowly increasing the distance between wedges and then slowly decreasing it.
4. The weight of hanger should always be included in the load.
5. The pulley should be free from friction.
6. The magnetization of the electromagnet should be checked with an iron needle before starting the experiment.
7.The electromagnet should be clamped 2-3 mm vertically above the center of the sonometer wire.
8. The sonometer wire should be made of steel so that it is attracted by the electromagnet

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