CP5a – Vibrating String: Effect of Length

Theory — Effect of Length on Frequency

In this experiment you keep tension T and mass per unit length μ constant and investigate how the resonant frequency f changes with vibrating length L.

What You Are Investigating

A string under tension, fixed at both ends, vibrates in a stationary wave pattern when driven at a resonant frequency. The simplest pattern (fundamental mode) has one loop. The resonant frequency depends on three things: the length L, the tension T, and the mass per unit length μ of the string.

In CP5a you change L only. T and μ stay fixed throughout.

The Key Relationship

From the fundamental equation f = (1/2L)√(T/μ), when T and μ are constant:

f ∝ 1/L i.e. f = k / L where k = √(T/μ)/2 is a constant

Doubling the length halves the frequency. Halving the length doubles the frequency.

Plotting f against 1/L gives a straight line through the origin. The gradient equals k = √(T/μ)/2, which equals half the wave speed on the string.

Stationary Waves and Harmonics

At the fundamental frequency (1st harmonic) there is one loop and λ = 2L. Higher harmonics occur at integer multiples of the fundamental:

1st harmonic: f₁ = v / 2L (1 loop, λ = 2L) 2nd harmonic: f₂ = v / L (2 loops, λ = L) 3rd harmonic: f₃ = 3v / 2L (3 loops, λ = 2L/3)

v = √(T/μ) = wave speed on the string · All harmonics satisfy the same f ∝ 1/L relationship.

The nodes (zero displacement) are at the fixed ends and at evenly spaced points along the string. The antinodes (maximum displacement) are midway between nodes.

Fixed Variables in This Experiment

Tension T — set by the hanging mass. Keep the same masses throughout every reading.
Mass per unit length μ — determined by the string material and diameter. Use the same string throughout. Do not swap strings mid-experiment.

Procedure

Two methods to find resonance. Select the one your school uses — or try both.

Equipment

Signal generator · Vibration generator (mechanical oscillator) · String · Pulley and clamp stand · Slotted masses and hanger · Metre rule · String of known μ (weigh a 1.00 m length and divide by length)

Fixed throughout: Tension T (same masses on hanger) and mass per unit length μ (same string). Varied: Vibrating length L.

Method A — Fixed Frequency

Set the signal generator to a fixed frequency. Then change the vibrating length L until the string resonates. Record the resonant length for several different fixed frequencies.

1

Set up the apparatus

Attach one end of the string to the vibration generator and pass it over the pulley. Hang the masses on the free end to create tension T. Measure and record T = Mg. This stays constant for every reading.

💡 Add a small pre-tension (at least the hanger alone) before measuring L — this removes any kinks.

2

Choose and set a fixed frequency

Set the signal generator to a chosen frequency (e.g. 120 Hz). Write it down. Do not change it during this set of readings. Switch the output ON.

💡 Frequencies between 80–400 Hz work well. Start at 120 Hz for string lengths of 0.30–0.80 m.

3

Adjust L until resonance

Move the bridge or clamp position to change the vibrating length L. Watch the string carefully — at resonance the amplitude suddenly increases and a clear loop pattern appears. The string appears almost stationary.

💡 If you see 2 loops instead of 1, you are at the 2nd harmonic. Increase L (or use half the frequency) to get back to the fundamental.

4

Fine-tune and measure L

Adjust L slowly around the resonant position to find the length where amplitude is greatest. Measure this length carefully with a metre rule from the vibration generator pin to the pulley. Record L.

💡 Repeat 3 times and take the mean L to reduce random uncertainty. Percentage uncertainty in L is your main source of error.

5

Change frequency and repeat

Set the signal generator to a new frequency. Find the new resonant length L. Repeat for at least 6 different frequencies across a wide range (e.g. 80, 120, 160, 200, 250, 320 Hz).

💡 A wider range of f (and therefore 1/L) values gives a more reliable gradient on your graph.

6

Plot f against 1/L

Calculate 1/L for each reading. Plot f (y-axis) against 1/L (x-axis). Draw a best-fit straight line through the origin. The gradient = √(T/μ)/2 = v/2.

💡 The line must pass through the origin — if it doesn't, check for a systematic error in your L measurements (e.g. measuring from the wrong reference point).

Method B — Sweep Frequency

Fix the vibrating length L at a set value. Then sweep the signal generator frequency upward from low until the string resonates. Record the resonant frequency for several different fixed lengths.

1

Set up the apparatus

Attach the string as before. Apply the hanging masses to set tension T. Clamp the string at a chosen length L (e.g. 0.50 m). Measure L carefully — this is your variable, so accuracy matters.

💡 Mark the clamp position with a pencil mark on the bench so L does not shift during the experiment.

2

Sweep frequency from low upward

Start the signal generator at a low frequency (e.g. 20 Hz) with the output ON. Slowly increase the frequency. Watch the string — at the resonant frequency the amplitude suddenly becomes much larger and a clear loop pattern appears.

💡 Sweep slowly. If you turn the dial too fast you will pass through resonance without noticing it.

3

Fine-tune to maximum amplitude

When a resonance appears, adjust the frequency in small steps (±1 Hz) to find the exact frequency where the amplitude is greatest. Read the frequency from the signal generator display. Record f and L.

💡 The resonance indicator in the simulation turns green when you are within ~1% of the true resonant frequency. In a real experiment, watch for the moment the string suddenly vibrates with much greater amplitude.

4

Note higher harmonics (optional)

Continue sweeping upward — you will find resonance again at f₂ ≈ 2f₁ (2 loops) and f₃ ≈ 3f₁ (3 loops). These all confirm the same f ∝ 1/L relationship and can be added to your graph.

💡 The 2nd harmonic is at exactly twice the fundamental frequency only if the string has uniform μ. Any deviation indicates non-uniformity.

5

Change L and sweep again

Move the clamp to a new length (e.g. 0.60 m). Repeat the frequency sweep from low to find the new resonant frequency. Repeat for at least 6 different lengths.

💡 Try lengths across a wide range — e.g. 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80 m. Shorter L gives higher f.

6

Plot f against 1/L

Calculate 1/L for each reading. Plot f (y-axis) against 1/L (x-axis). Draw a best-fit straight line through the origin. The gradient = √(T/μ)/2 = v/2.

💡 Method B typically gives smaller uncertainty in f than Method A gives in L, making it the preferred method for a precise result.

Fixed Variables

String type Medium nylon

μ (mass/length) 1.20 g/m

Tension T 2.94 N (300 g)

Wave speed v —

Variable — Length L

Vibrating length L 0.50 m

0.10 m1.00 m

Resonance Method

Harmonic

Resonance status

—

Resonant f

—

Length L

—

1/L

—

Harmonic n

—

Readings recorded: 0 / 6 minimum

      Amplitude response curve — ▲ marks current frequency relative to resonance
    

Results Data Table

Fixed: T = 2.94 N, μ = 1.20 g/m. Varied: length L. Derived quantities calculated automatically.

#	n harmonic	L / m	1/L / m⁻¹	f / Hz	f₁ = f/n / Hz	T / N	μ / g m⁻¹
No data yet — go to Simulation tab.

Graph & Analysis — f vs 1/L

Plot of resonant frequency f (y-axis) against 1/L (x-axis). A straight line through the origin confirms f ∝ 1/L. Gradient = √(T/μ)/2 = v/2.

Calculated wave speed

— m s⁻¹ from gradient × 2

Graph parameters

Gradient (= v/2)—

R²—

Expected v = √(T/μ)—

% difference—

Interpretation

Record at least 3 readings to see analysis.

Discussion Questions

Write your answers and reveal model answers when ready.

Question 1

Explain why the graph of f against 1/L passes through the origin, and what this tells you about the relationship between f and L.

The graph passes through the origin because f = (1/2L)√(T/μ) = k/L where k is a constant. When 1/L = 0 (i.e. L → ∞), f → 0, so the line must pass through the origin. This confirms that f is directly proportional to 1/L (equivalently, f is inversely proportional to L). The relationship f ∝ 1/L means that doubling the vibrating length exactly halves the resonant frequency.

Question 2

The gradient of the f vs 1/L graph equals v/2, where v is the wave speed on the string. Explain why, and show how you can use the gradient to calculate v.

From f = (1/2L)√(T/μ), rearranging: f = (v/2) × (1/L) where v = √(T/μ). This is of the form y = mx with x = 1/L and gradient m = v/2. So the wave speed v = 2 × gradient. Since T and μ are known, the expected wave speed is v = √(T/μ), which can be compared with the measured value to assess experimental accuracy.

Question 3

Explain the difference between the fixed-frequency method and the sweep-frequency method for finding resonance. State one advantage of each.

Fixed-frequency method: the signal generator is set to a constant frequency and the vibrating length L is adjusted until resonance occurs. The measured quantity is L. Advantage: no signal generator display is needed; L can be measured precisely with a ruler. Sweep-frequency method: the length L is fixed and the signal generator frequency is swept upward until resonance occurs. The measured quantity is f. Advantage: f can be read more precisely from a digital display than L from a ruler, reducing percentage uncertainty, especially for short string lengths.

Question 4

A student finds that their f vs 1/L graph does not pass through the origin — the y-intercept is positive. Suggest a reason for this systematic error.

A positive y-intercept means that as 1/L → 0 (L → ∞), f approaches a non-zero value, which is physically impossible. A likely cause is that the student measured L from the wrong reference point — for example, from the vibration generator body rather than the vibrating pin, or from the edge of the pulley rather than its centre. This adds a fixed length δL to every measurement, so the true vibrating length is L + δL rather than L. Using 1/(L + δL) instead of 1/L shifts all x-values to the left and produces a non-zero intercept.

Question 5

State two precautions a student should take to improve the reliability of this experiment.

1. Use the same string throughout — replacing the string changes μ, which changes the resonant frequency for a given L, introducing a systematic error. Measure μ accurately by weighing a 1.00 m length of the string before starting. 2. Measure L consistently from the same reference points — use the tip of the vibration generator pin to the centre of the pulley each time. Any inconsistency in reference points introduces a random or systematic error in L, which directly affects the gradient of the f vs 1/L graph.