# Vibrating Strings

## Apparatus:

Sonometer with a movable bridge, Magnadur magnets and yoke, 1kg mass
holder, long wires, two 0.5kg masses,

two 1kg masses, two 2kg masses, AFO audio frequency generator, two
crocodile clips, metre rule, frequency meter.

## Diagram:

## Procedure:

1. Set up the apparatus as shown in the diagram. The AFO may look
different.

Note the positions of the crocodile clips and the fact that the low
impedance / loudspeaker output of the audio frequency oscillator (AFO)
is used.

2. Initially, hang a mass, **m** = 3kg from the
sonometer wire in order to produce a tension, **T** = 30N
in the wire. Place the movable bridges so that length,
**L** is about 30cm. Measure this length and keep it
constant throughout the experiment.

3. Set the output of the AFO to 250Hz sinusoidal. The AFO acts as an
a.c. power supply causing an alternating current to flow along the
sonometer wire. Place the magnadur magnets half-way between the bridges.
With the magnets in place, the motor effect results in the wire
vibrating at the same frequency as the a.c., initially 250Hz.

4. Reduce the AFO frequency and find the lowest frequency that causes
the wire to resonate (usually around 200 Hz with a tension of 30N).

This is the fundamental frequency, **fo** of the wire
for the conditions chosen. Measure this frequency, **fo**
on the frequency meter (which you should connect across the AFO
output).

5. Repeat the measurement of **fo** by starting with a
frequency below your measurement above.

Hence find a mean value of **fo**.

6. Repeat stage three for six more masses, **m** between
1.0kg and 7.0kg, producing tensions between 10N and 70N.

7. At the resonant frequency, the length, **L** is equal
to half of the wavelength of the progressive waves that make up the
stationary wave on the wire.

Use the wave equation; **v = f** x **λ** to
calculate the speed of the progressive waves for all the cases above and
hence produce a results table that includes the above plus the square of
the wave speed.

8. According to theory, the wave speed is given by: **v =
√(T/µ)**.

Where **µ** is equal to the mass of a one metre length
of the wire.

Plot a graph of **(speed)**^{2} against
**tension**. It should be a straight line.

9. Measure the gradient of your graph and use its value to find
**µ** the mass per unit length of the sonometer wire.

10. **Answer the following questions:**

(a) Estimate the precision of the first value of **fo**
that you obtained.

(b) Calculate the percentage uncertainty of this value.

(c) Estimate the probable error in your value of
**µ**.