EXPERIMENT 6

THE SONOMETER

A traverse wave moving along a stretched string or flexible wire travels with a

velocity v given by

v = (1)

where T is the tension in the string and m is the mass per unit length of the string.

When a string is set vibrating, stationary waves can be set up due to reflection at

the ends. If both ends are fixed they are nodes or points of no vibration. The simplest

stationary wave is one with an antinode at the centre and with the length of string equal

to half a wavelength. This mode of vibration is termed the fundamental and is

illustrated. in Fig. 1. It may be seen that in the fundamental mode the wavelength is

related to the length L of the string by

L =

6-1

The string may also vibrate to produce overtones of harmonics as illustrated in

Fig. 2. Now the string vibrates in two or more segments with a node between

neighbouring segments.

In general, L and are related by

L = (2)

where n is the number of segments in the stationery wave. In particular n = 1 for the

fundamental mode.

The frequency is obtained from the wave relation v = and Eq (1) as

= (3)

The vibration of the string sets up longitudinal sound vibrations in the air

surrounding it and Eq. (3) represents the frequency of the note heard. Conversely, the

string may be set in resonant vibration by impulses having same frequency as one of its

modes of vibration. The impulse may be supplied by a vibrating tuning fork or, in the

case of conducting wires, by electromagnetic forces on an alternating current in a

magnetic field.

6-2

The sonometer consists of a long wooden sounding box as shown in Fig. 3. A

steel wire is fixed to a peg at one end of the box. It then passes over two movable

bridges C, C and a pulley, and is stretched by the application of a weight W. The

movable bridges determine the length L of vibrating strings and the tension in it is

given by the applied weight. To avoid accidents owing to the wire snapping the

weights applied should not exceed 5 kg wt.

Experiment (a) To verify the relation L for a constant stretching force

Apparatus

Sonometer, weights and hanger, set of tuning forks.

Procedure

Attach a hanger to the free end of the wire and apply weights so that the

stretching force totals 4.5 kg wt. Set the bridge C near the pegged end of the wire.

Start with the fork of the lowest frequency. Adjust the bridge C until the wire, when

plucked, gives a note near that of the fork. When this is achieved, those who have good

musical ears may hear beats when the fork and the wire are sounded together.

6-3

Next sound the fork and place its shank against the sonometer box. This should

set the wire to vibrate if its resonant frequency is the same as that of the fork. To detect

the vibration place a light paper rider (1 or 2 mm wide) on the wire between C and C.

C is moved slowly when the wire and fork are nearly in tune, the rider will jump up

and down or slide over the wire slightly. When a resonance is achieved, the rider will

jump so violently that it is thrown out from the wire. Note the vibrating length at

resonance.

To ascertain that this length corresponds to the fundamental mode, place the

rider at the midpoint of the resonant length. The rider would react most strongly at this

position. Repeat a number of times to get the best values for this length.

Determine the resonant lengths in tune with the other forks in succession and

tabulate your results. Plot a graph of against L . Knowing the tension (in

newtons) deduce the mass per unit length of the wire from the gradient and also the

velocity of the wave travelling along the string.

Experiment (b) Resonance in harmonics

Apparatus

Same as for (a).

Procedure

Use the tuning fork of the highest frequency. Double and triple the resonant

length of the wire obtained in (a) and try to achieve a unison. Determine the vibrating

lengths accurately. Compare these results with and 3/2, where is the

wavelength or this frequency as obtained in (a). Repeat the procedure for a different

tuning fork. Tabulate your results.

6-4

Experiment (c) Resonance with an alternating current

Apparatus

Sonometer, weights of 100 g to 700 g, mains transformer capable of delivering

2V A.C., a rheostat, a permanent bar magnet.

Procedure

Apply an A.C. voltage of 2V from a main transformer to the horizontal portion

of the wire of the sonometer through a rheostat in series with transformer. The

stretching weights to be used should be between 100 g and 700 g. A permanent bar

magnet is placed so that the central portion of the wire lies in the strong field near one

of its ends.

The directions of the magnetic field and the current are at right angles to each

other. The magnetic force on the current will be at right angles to both and will be

reversed in direction when the current is reversed. As the current is alternating, the

force on the wire will alternate in direction with the same frequency as the current. If

this is also the natural frequency of the wire under tension, resonance will take place.

This can be seen by the marked increase in the amplitude of vibration.

Starting with the smallest load available obtain the resonant lengths of the wire

for 4 or 5 different loads. Plot T vs L and deduce the frequency of the alternating

current.

Applying the smallest load obtain the first two resonant lengths and note in each

case the number of segments in which the vibration takes place. Explain your

observations with the help of diagrams.