A sonometer is a fascinating device used to explore the relationship between the frequency of sound produced by a string and various factors such as tension, length, and mass per unit length. Let’s delve into the details:

1. Construction of a Sonometer:

   • A sonometer consists of a hollow box that is approximately one meter long.
   • Attached to this box is a uniform metallic string.
   • The first end of the string is connected to a hook, while the second end is linked to a weight hanger via a pulley.
   • The sonometer typically uses a single string, making it akin to a monochord.
   • To adjust the tension in the string, weights are added at the free end.
   • Two adjustable knives are placed to modify the vibrating length of the stretched string.

2. Working Principle:

   • Once the setup is complete, a transverse standing wave forms at the edges of the knives, with nodes and antinodes.
   • Mathematically, we can express the relationship as follows:
     • Let (l) be the length of the vibrating element: (l = \frac{\lambda}{2}) (where (\lambda) represents the wavelength).
     • The frequency (f) of the vibrating element is related to the tension (T) and mass per unit length (\mu): [f = \frac{v}{\lambda} = \frac{l}{2l} \sqrt{\frac{T}{\mu}}]
     • The mass per unit length (\mu) can be calculated using the density (\rho) and diameter (d) of the string: [\mu = \frac{\pi r^2 \rho}{4} = \frac{\pi \rho d^2}{4}]
     • Thus, the frequency becomes: [f = \frac{l}{ld} \sqrt{\frac{T}{\pi \rho}}]

3. Laws of Transverse Vibrations:

   • The behavior of a stretched string in a sonometer follows three fundamental laws:
     • Law of Length: The frequency varies inversely with the resonating length, assuming constant mass per unit length and tension: (F \propto \frac{1}{l}).
     • Law of Tension: The frequency varies directly with the square root of tension, assuming constant resonating length and mass per unit length: (F \propto \sqrt{T}).
     • Law of Mass: The frequency varies inversely with the square root of mass per unit length, assuming constant length and tension: (F \propto \frac{1}{\sqrt{m}}).

4. Applications of Sonometer:

   • The sonometer is primarily used for demonstrating the relationships between frequency, tension, length, and mass per unit length of a string.
   • It serves as an educational tool in physics classrooms and laboratories.