1. Aim

To study the relation between frequency and length of a given wire under constant tension using a sonometer and verify that frequency is inversely proportional to the length of the vibrating wire.

2. Apparatus Used

• Sonometer with wire
• Set of weights/hanger
• Wooden bridges
• Tuning forks of different frequencies with rubber pad
• Rubber hammer
• Paper rider
• Meter scale
• Electromagnet (optional)

3. Diagram

Fig. 1: Experimental setup of the sonometer showing the wire, bridges, weight hanger, and tuning fork arrangement.

4. Theory

A sonometer is an apparatus used to study the vibrations of stretched strings. It consists of a hollow wooden box with a wire stretched over it. The wire is fixed at one end and passes over a pulley at the other end, where weights can be attached to vary the tension.

When a wire or string under tension vibrates, it produces a sound with a specific frequency. According to the laws of vibration of stretched strings, the frequency of vibration depends on:

• Length of the vibrating part of the string
• Tension in the string
• Mass per unit length (linear density) of the string

According to the theory of transverse vibrations in a stretched string, the fundamental frequency of vibration is given by:

For a given wire under constant tension, the linear density \( \mu \) remains constant. Therefore, the above equation reduces to:

This shows that the frequency of vibration is inversely proportional to the length of the vibrating string when the tension is kept constant.

5. Formula

6. Procedure

1. Set up the sonometer on a horizontal table. Ensure that the wire is properly stretched over the wooden box.
2. Attach a suitable weight to the free end of the wire passing over the pulley to maintain constant tension throughout the experiment.
3. Place two bridges on the sonometer board to define the vibrating length of the wire.
4. Select a tuning fork of known frequency (e.g., 256 Hz).
5. Strike the tuning fork gently with a rubber hammer and place its stem on the sonometer box.
6. Adjust the position of one of the bridges to vary the length of the vibrating wire until resonance occurs (when the wire vibrates with maximum amplitude).
7. To detect resonance, place a small paper rider on the wire. At resonance, the paper rider will be thrown off.
8. Measure the length of the wire between the two bridges using a meter scale.
9. Repeat the procedure with different tuning forks of various known frequencies.
10. Record all observations in the observation table.

7. Observation Table

Weight attached to the wire for tension: _______ kg

8. Calculations

For each observation, calculate the product of frequency and length (f × L).

Calculate the mean value of all products:

Alternatively, you can plot a graph of frequency (f) versus reciprocal of length (1/L) to verify the relation \( f \propto \frac{1}{L} \). This should give a straight line passing through the origin.

9. Result

From the experimental observations and calculations:

1. The mean value of the product of frequency and length (f × L) is found to be approximately constant (K_mean = _____ Hz-cm).
2. This verifies that the frequency of vibration of a stretched string is inversely proportional to its length when the tension is kept constant.
3. The relationship \( f \propto \frac{1}{L} \) or \( f \times L = \text{constant} \) is verified.

10. Precautions

1. The sonometer should be placed on a firm, horizontal surface to avoid any external vibrations.
2. The wire should be uniform and free from kinks or deformations.
3. The bridges should be sharp-edged and placed perpendicular to the wire.
4. The tension in the wire should remain constant throughout the experiment.
5. The tuning fork should be struck gently with a rubber hammer to produce pure sound.
6. The stem of the tuning fork should be placed firmly on the sonometer box.
7. The paper rider used to detect resonance should be very light and placed at the center of the wire.
8. Measurements of length should be taken accurately, ensuring that the zero error of the meter scale is accounted for.
9. Resonance should be carefully identified when the paper rider shows maximum displacement.
10. The experiment should be conducted in a quiet room to avoid interference from external sounds.

11. Sources of Error

1. Non-uniform cross-section of the wire can lead to variation in linear density.
2. Friction at the pulley may affect the actual tension in the wire.
3. Temperature variations can cause changes in the tension and properties of the wire.
4. The bridges may not be perfectly sharp or perpendicular to the wire.
5. Errors in measuring the exact length of the vibrating segment of the wire.
6. The tuning fork may not produce exact frequencies as marked on them.
7. External vibrations or sounds can interfere with the detection of resonance.
8. The non-rigidity of the sonometer box can absorb some vibrations.
9. Human error in identifying the exact moment of resonance.
10. Slippage of the wire over the bridges during vibrations.

12. Viva Voice Questions

A sonometer is an apparatus used to study the vibrations of stretched strings and to verify the laws of vibration. It works on the principle that a stretched string vibrates in resonance with a tuning fork when their frequencies match, and the frequency of vibration depends on the length, tension, and linear density of the string.

The frequency of a stretched string is inversely proportional to its length when tension and linear density are kept constant. Mathematically, \( f \propto \frac{1}{L} \) or \( f \times L = \text{constant} \).

If the tension (T) is doubled while keeping the length constant, the frequency increases by a factor of \( \sqrt{2} \). This is because \( f \propto \sqrt{T} \) when length and linear density are constant.

The frequency is inversely proportional to the square root of the mass per unit length (linear density). Mathematically, \( f \propto \frac{1}{\sqrt{\mu}} \) when length and tension are constant.

Resonance is the phenomenon where the wire vibrates with maximum amplitude when its natural frequency matches the frequency of the tuning fork. It is detected by placing a small paper rider on the wire, which gets thrown off at resonance due to the maximum amplitude of vibrations.

The tension must be kept constant to specifically study the relationship between frequency and length. If tension varies, it becomes an additional variable affecting the frequency, making it difficult to isolate the effect of length alone on the frequency.

The fundamental frequency is the lowest frequency at which a stretched string vibrates. It occurs when the string vibrates as a whole with nodes only at the ends, forming a single loop with maximum amplitude at the center.

Overtones are frequencies higher than the fundamental frequency at which a string can vibrate. Harmonics are specific overtones that are integer multiples of the fundamental frequency. The frequencies of harmonics are given by \( f_n = n \times f_1 \), where \( f_1 \) is the fundamental frequency and \( n \) is an integer.

The hollow wooden box acts as a resonator that amplifies the sound produced by the vibrating wire. The air inside the box vibrates in sympathy with the wire, increasing the intensity of the sound and making the resonance more easily detectable.

To study the relationship between frequency and tension, I would keep the length of the wire constant by fixing the positions of the bridges. Then, I would vary the tension by changing the weights attached to the wire and find the resonating length for each tension value. The frequency should be proportional to the square root of tension (\( f \propto \sqrt{T} \)).