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# Objective

• To study the relation between frequency and length of a given wire under constant tension using sonometer. To plot a graph between ν and l.
• To study the relation between length of a given wire and tension for constant frequency using sonometer. To plot a graph between l2 and T.

# Theory   What is a Sonometer ?

Sonometer consists of a hollow rectangular wooden box of more than one meter length, with a hook at one end and a pulley at the other end. One end of a string is fixed at the hook and the other end passes over the pulley. A weight hanger is attached to the free end of the string. Two adjustable wooden bridges are put over the board, so that the length of string can be adjusted.

## Production of transverse waves in stretched strings

If a string which is stretched between two fixed points is plucked at its center, vibrations produced and it move out in opposite directions along the string. Because of this, a transverse wave travels along the string.

If a string of length l having mass per unit length m is stretched with a tension T, the fundamental frequency of vibration f is given by;

## Laws of transverse vibrations on a stretched string:

• Law of Length: The frequency of vibration of a stretched string varies inversely as its resonating length (provided its mass per unit length and tension remain constant.)

• Law of Tension:The frequency of vibration of a stretched string varies directly as the square root of its tension, (provided its resonating length and mass per unit length of the wire remains constant).

​Relation between frequency and length

From the law of length,  f ×l = constant

A graph between f and 1/l will be a straight line.

### Relation between length and tension

From the equation for frequency, √T / l = constant

A graph between T and l2 will be a straight line.

# Learning outcomes

• Students develop the idea about standing waves.
• Students understand the sonometer apparatus and its working.
• Students get the relation between frequency, length and tension of a stretched string under vibration.

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# LABAPPARA Materials Required

• A sonometer
• A set of tuning forks of known frequency
• 0.5kg weight hanger
• Some 0.5kg slotted weights
• Paper rider

# Real Lab Procedure

## To find the relation between frequency and length

• Place the sonometer on the table.
• Make sure that the pulley is frictionless. If you feel any friction, oil them.
• Stretch the wire by placing a suitable maximum load on the weight hanger.
• Move the wooden bridges outward, so that the length of wire between the bridges is maximum.
• Take a tuning fork of known frequency. Make it vibrate by strike its prong with a rubber pad. Bring it near the ear.
• Pluck the sonometer wire and leave it to vibrate.
• Compare the sounds produced by tuning fork and sonometer wire. (Sound which has low pitch has less frequency).
• Gently adjust the bridges for decreasing the length of wire, till the two sounds appear alike.
• Put an inverted V shaped paper rider on the middle of the wire.
• Vibrate the tuning fork and touch the lower end of its handle with sonometer board. The wire vibrates due to resonance and the paper rider falls.
• Measure the length of wire between the bridges using a meter scale. It is the resonant length and record it in the ‘length decreasing’ column.
• Now, bring the bridges closer and then slowly increase the length of the wire till the paper rider falls.
• Measure the length of wire and record it in ‘length increasing’ column.
• Repeat the above steps with tuning forks of other frequencies, and find resonant length each time.

## To find the relation between length and tension

• Select a tuning fork of known frequency
• Set the load in the weight hanger as maximum.
• Repeat the steps in the previous section to find out the resonant length.
• Now, remove 0.5kg weight from the weight hanger and find resonant length with same tuning fork.
• Repeat the experiment by removing slotted weights one by one in equal steps of 0.5kg.
• Record the observations each time.

# Procedure

• Select the environment from the drop down list.
• Select the material of the wire from the drop down list.
• Select the diameter of the wire using the slider.
• Select the weight of the slotted weights using the slider.
• Select the frequency of the tuning fork using the slider.
• Click on the ‘Hit tuning fork’ button to start/stop the vibration of tuning fork and touch it with the sonometer board.
• Change the position of bridge A using the slider.
• Change the position of bridge B using the slider.
• Click on the ‘Place the paper rider’ button to place the paper rider back.
• To redo the experiment, click on the ‘Reset’ button.

# Observations

## To find the relation between frequency and length

Constant tension on the wire, T= .........kg
 Sl No. Frequency of tuning fork used, f (Hz) Resonant length of wire 1/ l (cm-1) Length increasing l1(cm) Length decrasing l2 (cm) Mean                 l = (l1 +l2) / 2

## To find the relation between length and tension

 Sl No. Load, M (kg) Tension, T=Mg (N) Resonant length of wire l 2 (cm2) l 2 / T (cm2 / N) Length increasing l1(cm) Length decrasing l2 (cm) Mean           l = (l1 +l2) / 2
Mean, l2 / T =.................. cm2 / N

# To find the relation between frequency and length

• Find mean resonant length, l
• Calculate 1/l in each case.
• Plot a graph between frequency and reciprocal of length, taking frequency along X axis and reciprocal length along Y axis.

## To find the relation between length and tension

• Find square of resonant length (l2) each time.
• Calculate corresponding l2/T value.
• Plot a graph between square of length and tension, taking tension along X axis and square of length along Y axis.

# Results:

The frequency V/s reciprocal of length graph is a straight line, which indicates that, frequency is inversely proportional to resonant length. From the tabular column, it is found that; l2/T is a constant. The graph between square of length and tension is a straight line, which shows that tension is directly proportional to square of resonant length.