Sonometer

Construction

A sonometer consists of a hollow rectangular wooden box called the sound box. The sound box makes a larger mass of air vibrate, so the sound produced by the vibrating metal wire is amplified; the same principle is used in stringed instruments such as the violin, the guitar and the tanpura.

Two bridges P and Q run across the width of the box and can be moved parallel to its length. A metal wire of uniform cross-section runs along the length of the box over these bridges. It is fixed at one end, and its other end passes over a pulley and carries a hanger to which slotted weights can be added. Changing the weights changes the tension in the wire; moving the bridges changes the vibrating length AB.

Working

If the wire is plucked at a point midway between the bridges, transverse waves are produced. They are reflected at the bridges and superpose, so stationary waves are set up between the two bridges; the portion AB of the wire between P and Q is the vibrating length.

The wire can also be made to vibrate by holding a vibrating tuning fork near it, or by holding the stem of the fork against the sonometer box. The frequency of vibration is then the same as that of the fork. If this frequency happens to be one of the natural frequencies of the wire, standing waves of large amplitude are set up because the two vibrate in resonance.

To identify resonance, a small piece of paper called the rider R is placed over the wire midway between the bridges. When the frequency of the tuning fork matches the frequency of the fundamental mode of the wire — achieved by adjusting the length AB with the bridges P and Q — the rider sits at an antinode and is thrown off the wire.

Diagram in Words

Draw a long hollow rectangular box seen from the side. A wire runs along the top, fixed to a peg at the left end, passing over two inverted-V bridges P and Q, then over a pulley clamped at the right edge of the table, carrying a hanger with slotted weights. Mark the vibrating length AB between the bridges and place a small paper rider R at the midpoint of AB. Label the box as the sound box.

Verification of the Laws of a Vibrating String

First law — the law of length

By measuring the length of the wire and its mass, the mass per unit length $m$ is determined. The wire is then stretched on the sonometer and the hanger is suspended from its free end. A suitable tension $T$ is applied by placing slotted weights on the hanger.

A light paper rider is placed on the wire midway between the bridges. The tuning fork is set into vibration by striking it on a rubber pad, and its stem is held in contact with the sonometer box. By changing the distance between the bridges without disturbing the rider, the frequency of vibration of the wire is changed. When the frequency of the wire becomes exactly equal to that of the fork, the wire vibrates with maximum amplitude and the paper rider is thrown off; that length is noted.

A set of tuning forks of different frequencies $n_1, n_2, n_3,\ldots$ is used, and the corresponding vibrating lengths $l_1, l_2, l_3,\ldots$ are noted, keeping the tension $T$ constant. It is observed that

$$n_1l_1 = n_2l_2 = n_3l_3 = \ldots = \text{constant}$$

$$\therefore\ nl = \text{constant} \qquad \text{i.e.}\quad n \propto \dfrac{1}{l} \quad \text{if } T \text{ and } m \text{ are constant}$$

Thus the first law of a vibrating string is verified.

Second law — the law of tension

The vibrating length $l$ of the given wire of mass per unit length $m$ is kept constant. By changing the tension, the same length is made to vibrate in unison with different tuning forks of various frequencies. If tensions $T_1, T_2, T_3,\ldots$ correspond to frequencies $n_1, n_2, n_3,\ldots$ it is observed that

$$\dfrac{n_1}{\sqrt{T_1}} = \dfrac{n_2}{\sqrt{T_2}} = \dfrac{n_3}{\sqrt{T_3}} = \ldots = \text{constant}$$

$$\therefore\ \dfrac{n}{\sqrt{T}} = \text{constant} \qquad \text{i.e.}\quad n \propto \sqrt{T} \quad \text{if } l \text{ and } m \text{ are constant}$$

Third law — the law of linear density

Two wires of different mass per unit length $m_1$ and $m_2$ are used. The first wire is given a suitable tension and made to vibrate in unison with a given tuning fork; its vibrating length $l_1$ is noted. Using the same fork, the second wire is made to vibrate under the same tension and its vibrating length $l_2$ is determined. The frequency of vibration of the two wires is thus the same, under the same applied tension $T$. It is found that

$$l_1\sqrt{m_1} = l_2\sqrt{m_2} \qquad \therefore\ l\sqrt{m} = \text{constant}$$

But by the first law $n \propto \dfrac{1}{l}$, therefore

$$n \propto \dfrac{1}{\sqrt{m}} \quad \text{if } T \text{ and } l \text{ are constant}$$

In this way all three laws of a vibrating string are verified using a sonometer.

Load Immersed in Water

If the load stretching the wire has specific gravity $\rho$ and is completely immersed in water, the tension falls from $w_1$ to the apparent weight $w_2$, with $\dfrac{w_1}{w_1 - w_2} = \rho$, so $\dfrac{w_1}{w_2} = \dfrac{\rho}{\rho - 1}$. Since $n \propto \sqrt{T}$,

$$\dfrac{n_1}{n_2} = \sqrt{\dfrac{w_1}{w_2}} = \sqrt{\dfrac{\rho}{\rho - 1}}$$

The frequency falls, so the beats heard with the fork give $n_1 - n_2$.

Standard Results

• Resonance is detected by the paper rider flying off the wire.
• Law of length: $nl =$ constant. Law of tension: $\dfrac{n}{\sqrt{T}} =$ constant. Law of linear density: $l\sqrt{m} =$ constant.
• $n = \dfrac{1}{2l}\sqrt{\dfrac{T}{m}}$, with $T = Mg$ for a hanging mass $M$ and $g = 9.8$ m s$^{-2}$.
• Load of specific gravity $\rho$ immersed in water: $\dfrac{n_1}{n_2} = \sqrt{\dfrac{\rho}{\rho - 1}}$.