To verify the laws of vibration of stretched strings using a sonometer and to study the relationship between the length of a given wire and tension for constant frequency.

• Sonometer with wire
• Set of weights/hanger
• Meter scale
• Tuning fork of known frequency
• Rubber pad
• Paper rider
• Wooden bridges
• Rubber hammer
• Weight box

The sonometer is an apparatus used to study the laws of vibration of stretched strings. When a stretched string of length L is plucked, it vibrates with its fundamental frequency which depends on:

• The length of the vibrating part of the string (L)
• The tension in the string (T)
• The mass per unit length of the string (μ)

According to the laws of vibration of stretched strings, the fundamental frequency of vibration (f) is given by:

For a given wire with constant mass per unit length (μ) and at constant frequency (f), we can derive:

Or equivalently:

This means that for a given wire vibrating with constant frequency, the square of its length is directly proportional to the tension applied to it. Alternatively, we can write:

In this experiment, we will verify this relationship by adjusting the length and tension of the sonometer wire to match the frequency of a standard tuning fork.

1. Set up the sonometer on a horizontal table and ensure it is stable.
2. Measure the diameter of the wire using a screw gauge at different positions and calculate the average.
3. Calculate the mass per unit length (μ) of the wire if given, or use the standard value provided.
4. Place two wooden bridges on the sonometer box at a convenient distance (approx. 60-70 cm apart).
5. Hang the weight hanger at one end of the sonometer wire passing over the pulley.
6. Add an initial weight (e.g., 0.5 kg) to the hanger.
7. Place a small paper rider at the middle of the wire between the bridges.
8. Strike the tuning fork on a rubber pad and hold it near one of the bridges.
9. Adjust the position of one of the bridges until the paper rider falls off, indicating resonance (the wire and tuning fork have the same frequency).
10. Measure and record the length of the wire between the two bridges.
11. Increase the weight on the hanger in steps (e.g., 0.5 kg each time).
12. For each weight (tension), repeat steps 7-10 to find the new resonating length.
13. Continue until you have at least 6-8 readings.
14. Calculate L² for each length and plot a graph of L² vs T.

Tuning fork frequency (f) = ______ Hz

Diameter of the wire (d) = ______ mm

Mass per unit length of the wire (μ) = ______ kg/m

S.No.	Mass (M) kg	Tension (T=Mg) N	Resonating Length (L) m	L² (m²)	L²/T (m²/N)
1
2
3
4
5
6

Step 1: Calculate the tension in the wire for each mass:

Step 2: Calculate L² for each measured length.

Step 3: Calculate L²/T for each observation to verify if it remains constant.

Step 4: Plot a graph of L² (y-axis) versus T (x-axis).

Step 5: Calculate the slope (K) of the graph:

Step 6: Using the calculated slope and known frequency of the tuning fork, calculate the mass per unit length of the wire:

Step 7: Compare the calculated value of μ with the given value to determine the percentage error:

1. The graph between L² and T is a straight line passing through the origin, verifying that L² ∝ T.
2. The value of L²/T = ______ m²/N (average value).
3. The calculated mass per unit length of the wire (μ) = ______ kg/m.
4. The percentage error in the calculated value of μ = ______ %.
5. The laws of vibration of stretched strings are verified.

1. The sonometer should be placed on a horizontal table to avoid any tilt.
2. The wire should be uniform, without any kinks or defects.
3. The bridges should be sharp-edged and perpendicular to the wire.
4. The paper rider should be very light and placed at the middle of the wire.
5. The tuning fork should be struck gently on a rubber pad, not too hard.
6. The resonance position should be approached from both sides to determine the exact point.
7. Ensure there is no slack in the wire when taking measurements.
8. The tuning fork should be held near one of the bridges, not touching it.
9. The weights should be added carefully to avoid jerks.
10. Read the length measurement carefully, with the eye perpendicular to the scale.

1. The wire may not be perfectly uniform throughout its length.
2. The bridges may not be perfectly sharp-edged or perpendicular to the wire.
3. The pulley may have friction, affecting the actual tension in the wire.
4. Temperature variations during the experiment can affect the tension in the wire.
5. The tuning fork's frequency might vary slightly if not struck properly.
6. Errors in identifying the exact resonance position.
7. Parallax errors while reading the length measurements.
8. The effect of the weight of the wire itself, which affects the actual tension.
9. Air currents may affect the vibration of the wire.
10. Inaccuracies in the weights used for applying tension.