1. AIM

To determine the frequency of AC mains using a sonometer and an electromagnet.

2. APPARATUS USED

• Sonometer with a steel wire
• Electromagnet (with soft iron core)
• Step-down transformer
• Weights/Hanger with slotted weights
• Meter scale
• Paper rider
• AC mains supply
• Connecting wires
• Wooden bridges (2)
• Vernier caliper
• Screw gauge
• Knife edge

3. DIAGRAM

4. THEORY

The sonometer is an apparatus used to study the laws of vibration of stretched strings and to determine frequency. It consists of a hollow wooden resonance box with a wire stretched over it. The wire passes over two fixed bridges at the ends and a movable bridge in between. The wire is kept taut using weights attached to one end.

Principle:

When an alternating current from the AC mains passes through an electromagnet placed beneath a sonometer wire, the wire experiences a periodic force due to the alternating magnetic field. This causes the wire to vibrate with the same frequency as that of the AC mains.

If the tension in the wire is adjusted such that its natural frequency of vibration becomes equal to the frequency of the AC mains, resonance occurs. At resonance, the wire vibrates with maximum amplitude.

For a stretched string:

The fundamental frequency of vibration of a stretched string is given by:

Where:

• \(f\) = frequency of vibration (Hz)
• \(l\) = length of the vibrating wire (m)
• \(T\) = tension in the wire (N)
• \(m\) = mass per unit length of the wire (kg/m)

When the AC electromagnet is placed beneath the sonometer wire, the wire is forced to vibrate at the frequency of AC mains. When the natural frequency of the wire equals the frequency of AC mains, resonance occurs.

At resonance:

The mass per unit length (m) can be calculated as:

Where:

• \(M\) = total mass of the wire (kg)
• \(L\) = total length of the wire (m)

Alternatively, if the diameter of the wire is known:

Where:

• \(d\) = diameter of the wire (m)
• \(\rho\) = density of the material of the wire (kg/m³)

For a given wire, if we vary the tension and find the corresponding resonant length, we get:

Or for a fixed tension, if we vary the length:

5. FORMULA

The frequency of AC mains is calculated using the formula:

Where:

• \(f_{AC}\) = frequency of AC mains (Hz)
• \(l\) = resonant length of the wire (m)
• \(T\) = tension in the wire (N)
• \(m\) = mass per unit length of the wire (kg/m)

The tension in the wire is given by:

Where:

• \(M\) = mass suspended (kg)
• \(g\) = acceleration due to gravity (9.8 m/s²)

The mass per unit length can be calculated as:

Where:

• \(d\) = diameter of the wire (m)
• \(\rho\) = density of the material of the wire (kg/m³)

Alternatively, if the total mass and length of the wire are known:

6. PROCEDURE

1. Setup Preparation:

   • Place the sonometer on a horizontal table.
   • Measure the diameter of the wire using a screw gauge at several places and take the average.
   • Fix the wire on the sonometer passing it over the two fixed bridges at the ends.
   • Suspend a suitable weight (e.g., 1 kg) to create tension in the wire.

2. Electromagnet Setup:

   • Connect the electromagnet to the AC mains through a step-down transformer.
   • Place the electromagnet below the sonometer wire without touching it.

3. Finding Resonance:

   • Place two bridges on the sonometer box at a suitable distance apart (e.g., 50-60 cm).
   • Place a small paper rider on the wire between the bridges.
   • Switch on the AC supply to the electromagnet.
   • Adjust the position of one of the bridges slowly until the paper rider vibrates with maximum amplitude and falls off. This indicates resonance.
   • Measure the length of the wire between the two bridges precisely using a meter scale.

4. Variation of Length with Tension:

   • Keep the electromagnet position fixed.
   • Change the suspended weight to vary the tension in the wire.
   • For each tension, find the resonant length as described above.
   • Record the mass suspended and the corresponding resonant length in the observation table.
   • Take at least 5-6 observations by varying the suspended mass.

5. Verification (Optional):

   • Plot a graph of \(l^2\) versus \(T\).
   • The graph should be a straight line, confirming the relation \(l^2 \propto T\).

7. OBSERVATION TABLE

Measurement of Wire Parameters:

Measurement of Wire Diameter using Screw Gauge
Observation No.	Reading at Position 1 (mm)	Reading at Position 2 (mm)	Reading at Position 3 (mm)	Reading at Position 4 (mm)	Mean Diameter (mm)
1

Calculation of Mass per Unit Length:

Diameter of Wire, d (m)	Density of Wire Material, ρ (kg/m³)	Mass per Unit Length, m (kg/m)

Resonance Observations:

S.No.	Suspended Mass, M (kg)	Tension, T = Mg (N)	Resonant Length, l (m)	l² (m²)	Frequency of AC Mains, f = (1/2l)√(T/m) (Hz)
1
2
3
4
5
6

Note: The value of g used in calculations is 9.8 m/s².

8. CALCULATIONS

1. Calculation of mass per unit length (m):
   \[ m = \frac{\pi d^2 \rho}{4} \]

   Given:

   • Diameter of wire, d = _____ m
   • Density of steel wire, ρ = _____ kg/m³ (typical value for steel: 7800 kg/m³)

   Substituting the values:

   \[ m = \frac{\pi \times (_____)^2 \times _____}{4} = _____ \text{ kg/m} \]
2. Calculation of tension (T) for each observation:
   \[ T = M \times g \]

   For Observation 1:

   • Suspended mass, M = _____ kg
   • Acceleration due to gravity, g = 9.8 m/s²

   Tension, T = _____ × 9.8 = _____ N

3. Calculation of frequency for each observation:
   \[ f = \frac{1}{2l}\sqrt{\frac{T}{m}} \]

   For Observation 1:

   • Resonant length, l = _____ m
   • Tension, T = _____ N
   • Mass per unit length, m = _____ kg/m
   \[ f = \frac{1}{2 \times _____}\sqrt{\frac{_____}{_____}} = _____ \text{ Hz} \]
4. Mean frequency of AC mains:
   \[ f_{mean} = \frac{f_1 + f_2 + f_3 + ... + f_n}{n} \]

   Where f₁, f₂, f₃, ... fn are the frequencies calculated from each observation, and n is the number of observations.

   \[ f_{mean} = \frac{_____ + _____ + _____ + _____ + _____ + _____}{6} = _____ \text{ Hz} \]

9. RESULT

The frequency of AC mains determined using the sonometer is _____ Hz.

The accepted standard value of AC mains frequency in your country is:

• 50 Hz (in most countries including India, Europe, Australia, etc.)
• 60 Hz (in countries like USA, Canada, parts of South America, etc.)

The percentage error in the measurement is:

10. PRECAUTIONS

1. The sonometer should be placed on a horizontal, rigid, and vibration-free surface.
2. The wire should be uniform and free from kinks or defects.
3. The bridges should be sharp-edged and perpendicular to the wire.
4. The electromagnet should be positioned directly below the wire without touching it.
5. The paper rider should be very light to detect resonance accurately.
6. The weights used for tension should be calibrated and accurate.
7. The length of the wire should be measured precisely, taking into account the effective vibrating length.
8. The current through the electromagnet should not be too high to avoid heating effects.
9. Avoid touching or disturbing the sonometer setup during measurements.
10. Take multiple readings and use the average to minimize random errors.
11. Ensure that the wire is well-tensioned and does not slip over the pulley.
12. The diameter of the wire should be measured at multiple points using a screw gauge to get an accurate average.

11. SOURCES OF ERROR

Instrumental Errors:

• Error in measuring the diameter of the wire using the screw gauge.
• Error in the calibration of weights used for tension.
• Limitations in the precision of the meter scale used to measure length.
• Non-uniform cross-section of the wire.
• Error in positioning the bridges exactly at nodes.

Observational Errors:

• Difficulty in precisely identifying the point of maximum resonance.
• Parallax error while taking length measurements.
• Subjective judgment in determining when the paper rider falls off.

Environmental and Physical Factors:

• Temperature variations affecting the tension and elasticity of the wire.
• Friction at the pulley affecting the actual tension in the wire.
• External vibrations interfering with the experiment.
• Air currents affecting the behavior of the paper rider.
• The electromagnet may induce non-uniform magnetic field across the wire length.
• Heating of the wire due to current-induced by the electromagnet.

Theoretical Limitations:

• The assumption that the wire is perfectly flexible and has no stiffness.
• Neglecting the effect of the mass of the wire itself on tension.
• The formula assumes ideal conditions of harmonic motion.

12. VIVA VOCE QUESTIONS