Sonometer

Sonometer Experiment for Schools, Teachers, and Students

A sonometer is a device used to study the transverse vibrations of stretched strings and verify the relationship between frequency, tension, length, and linear density of the string.

Theory:

1. Principle of Sonometer:

The sonometer works on the principle of resonance. When a vibrating tuning fork is placed near a stretched string, the string vibrates with maximum amplitude (resonance) if its natural frequency matches the frequency of the tuning fork.

2. Fundamental Frequency of a Vibrating String:

The frequency (f) of transverse vibrations in a stretched string is given by:

f = 1 / (2L) √(T / μ)

where:

• L: Length of the vibrating part of the string (in meters),
• T: Tension in the string (in Newtons),
• μ: Linear mass density of the string (μ = m / L, mass per unit length, in kg/m).

3. Factors Affecting Frequency:

• Length of the String (L):
  The frequency is inversely proportional to the length.
  f ∝ 1 / L
• Tension in the String (T):
  The frequency is directly proportional to the square root of the tension.
  f ∝ √T
• Linear Mass Density (μ):
  The frequency is inversely proportional to the square root of the linear density.
  f ∝ 1 / √μ

4. Laws of Vibrating Strings:

• Law of Length:
  For a given tension and mass per unit length, the frequency is inversely proportional to the length of the string:
  f ∝ 1 / L
• Law of Tension:
  For a given length and mass per unit length, the frequency is proportional to the square root of the tension:
  f ∝ √T
• Law of Mass:
  For a given tension and length, the frequency is inversely proportional to the square root of the linear density:
  f ∝ 1 / √μ

Procedure to Use a Sonometer:

• A wire is stretched over a sonometer box and fixed at both ends.
• A known mass (m) is suspended at one end of the wire to create tension (T = mg).
• A vibrating tuning fork of known frequency is brought close to the wire.
• By adjusting the length of the vibrating part of the string using a movable bridge, resonance occurs when the string vibrates with maximum amplitude.
• The vibrating length (L) is measured, and the frequency is calculated using the formula.

Applications of Sonometer:

• To determine the frequency of a tuning fork.
• To verify the laws of vibrating strings.
• To determine the linear mass density of the string.
• To find the speed of a transverse wave on the string.

Real-Life Examples:

• Musical instruments like guitars and violins use stretched strings to produce sound, which follow similar principles.
• Sonometers are used in laboratories for acoustic studies.
• Engineers use sonometers to measure tension in cables and wires.

Observations:

• The frequency increases as the tension in the string increases.
• The frequency decreases when the length of the vibrating string increases.
• For a given length and tension, a thinner (low-density) string produces a higher frequency.
• Resonance occurs only when the natural frequency of the string matches the external frequency (tuning fork).