16.1 – Verifying the Stretching of a Spring (Tension Model) Revision Notes for Leaving Cert Physics

16.1 – Verifying the Stretching of a Spring (Tension Model)

Purpose of the experiment

This experiment aims to verify the stretched string model for a vibrating wire. The mathematical relationship we're testing shows how the frequency of vibration depends on the physical properties of the string. This is fundamental to understanding how musical instruments work and how waves travel through different materials.

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This experiment demonstrates one of the most important principles in wave physics and acoustics. The relationship you'll verify is the same principle that governs how guitars, violins, and other stringed instruments produce different pitches.

The stretched string model

The model we're verifying states that the frequency of a vibrating string is given by:

$f = \frac{1}{2l}\sqrt{\frac{T}{\mu}}$

Where:

f = frequency of vibration (Hz)
l = length of the vibrating section (m)
T = tension in the string (N)
μ = linear mass density (kg m⁻¹)

This equation tells us that frequency increases with higher tension but decreases with greater length and mass per unit length.

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Understanding this relationship is crucial:

Higher tension → Higher frequency (tighter strings sound higher)
Longer length → Lower frequency (longer strings sound lower)
Greater mass density → Lower frequency (thicker strings sound lower)

Equipment needed

To conduct this experiment, you'll need:

A set of tuning forks with known frequencies
A sonometer (wire-stretching apparatus) with a newton balance
A metre stick for measuring lengths
A weighing scale for measuring wire mass

The sonometer allows you to adjust both the tension and the vibrating length of the wire, making it perfect for testing the mathematical model.

Experimental method

Setting up the apparatus

Prepare the wire: Make the wire as long as possible by adjusting the bridges on the sonometer. This gives you the maximum range for length measurements.
Start with low frequency: Choose the tuning fork with the lowest frequency available. Strike it and place its stem on one of the bridges to transfer vibrations to the wire.
Find resonance: Gradually increase the tension in the wire using the newton balance until the wire resonates with the tuning fork. You'll know resonance has occurred when the wire vibrates strongly, making the tuning fork sound much louder.

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Detecting Resonance: You'll hear a significant increase in volume when resonance occurs. The wire will also vibrate visibly, and you may feel vibrations through the sonometer frame.

Taking measurements

Measure the vibrating length: Use the metre stick to measure the distance between the two bridges. This is your length value (l).
Record the tension: Read the tension directly from the newton balance. This gives you the tension value (T).
Note the frequency: Record the frequency of the tuning fork you're using.

Collecting data

Repeat this process using different tuning forks with progressively higher frequencies. You may need to shorten the length of the string to achieve resonance with higher frequency tuning forks.

Determining linear mass density

If the linear mass density (μ) of the wire isn't already known, you'll need to measure it:

Carefully remove the wire from the sonometer
Measure its total mass using a weighing scale
Measure its total length
Calculate: $\mu = \frac{\text{total mass of wire}}{\text{total length of wire}}$

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Worked Example: Calculating Linear Mass Density

If a wire has:

Total mass = 0.0045 kg
Total length = 1.2 m

Then: $\mu = \frac{0.0045 \text{ kg}}{1.2 \text{ m}} = 0.00375 \text{ kg m}^{-1}$

Results and analysis

Once you've collected your data, substitute the values for l, T, and μ into the stretched string model:

$f = \frac{1}{2l}\sqrt{\frac{T}{\mu}}$

Calculate the theoretical frequency for each set of measurements. If the model is accurate, your calculated values should closely match the actual frequencies of the tuning forks you used.

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Worked Example: Verifying the Model

Given measurements:

Length (l) = 0.8 m
Tension (T) = 25 N
Linear mass density (μ) = 0.00375 kg m⁻¹

Step 1: Calculate the theoretical frequency $f = \frac{1}{2 \times 0.8}\sqrt{\frac{25}{0.00375}}$

Step 2: Simplify the calculation
$f = \frac{1}{1.6}\sqrt{6666.7} = \frac{81.6}{1.6} = 51.0 \text{ Hz}$

Step 3: Compare with tuning fork frequency If the tuning fork frequency was 50 Hz, the percentage error would be: $\text{Percentage error} = \frac{|51.0 - 50.0|}{50.0} \times 100\% = 2.0\%$

The closer your calculated frequencies are to the actual tuning fork frequencies, the better the model describes the behaviour of the stretched string.

Sources of experimental error

Understanding potential errors helps improve your experimental technique:

Random errors

Measurement uncertainty: Using a metre stick introduces small measurement variations
Parallax errors: Ensure you read measurements directly perpendicular to the scale
Resonance detection: It can be challenging to identify the exact point of maximum resonance

Systematic errors

Newton balance calibration: Ensure the balance reads zero when no tension is applied
String tension variations: Avoid applying tension when the newton balance shows zero, as this affects accuracy

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Common Mistake to Avoid: Never apply additional tension to the wire when the newton balance reads zero. This creates a systematic error by adding unmeasured tension to your readings.

Minimising errors

Take multiple measurements and calculate averages
Ensure all equipment is properly calibrated before use
Be consistent in your measurement technique throughout the experiment

Safety considerations

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⚠️ SAFETY WARNING: When working with a sonometer, the wire is under considerable tension and can snap unexpectedly. Always wear safety glasses when near a sonometer wire under tension to protect your eyes from potential wire fragments.

Key questions for understanding

Why is the piece of paper placed at the centre of the wire? This helps visualise the wire's vibration and makes it easier to detect resonance.
What type of error occurs when measuring distance between bridges? This would typically be a random error due to measurement limitations.
How can systematic errors be minimised? Through proper calibration of equipment and consistent measurement techniques.
What happens when tension increases? Higher tension leads to higher frequency vibrations, as shown in the mathematical model.

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Key Points to Remember:

The stretched string model $f = \frac{1}{2l}\sqrt{\frac{T}{\mu}}$ shows that frequency depends on length, tension, and mass density
Resonance occurs when the wire vibrates at the same frequency as the tuning fork
Always wear safety glasses when working with tensioned wires
Both random and systematic errors can affect your results - proper technique minimises these
The experiment verifies a fundamental principle used in musical instruments and wave physics
Higher tension and shorter length both increase frequency
Proper calibration and consistent measurement technique are essential for accurate results

16.1 – Verifying the Stretching of a Spring (Tension Model) (Leaving Cert Physics): Revision Notes