187; CP16OscillationsCore

Core Practical 16: Unknown Mass from Resonance

A resonance calibration turns measured frequencies into an unknown mass. Use the same oscillator and attachments for calibration and measurement so fixed extra masses remain represented.

40 min 13.1 Simple Harmonic Motion

What you'll be able to do

Find resonant frequencies for known masses
Construct a linear calibration for an unknown mass
Evaluate damping, effective spring mass and resolution

Choose and identify the resonance

Use the centre’s approved driven mass–spring oscillator. Attach a known load, vary the driving frequency and record the frequency giving maximum steady amplitude. Approach the peak from both sides with sufficiently small frequency steps; allow transients to settle before judging amplitude.

In light damping, the displacement-amplitude resonance is close to the natural frequency. Substantial damping broadens the peak and can shift its maximum, so an assumed equality requires qualification. Keep the spring, driving amplitude, attachments and geometry consistent, and avoid motion that stretches the spring beyond its elastic range.

Build a calibration with known masses

For an ideal mass–spring oscillator, f=(1/2π)√(k/m). With a fixed effective attachment/spring mass m0, the measured system has 1/f²=(4π²/k)(M+m0), where M is the added load mass. Plot 1/f² vertically against known added mass horizontally.

Fit a straight line over the useful range and retain a physically justified intercept; it can represent fixed moving mass. The same calibration can infer the unknown added mass from its measured frequency by interpolation. If total oscillating mass is used instead, define which fixed masses have already been included so they are not added twice.

\frac1{f^2}=\frac{4\pi^2}{k}(M+m_0)

Infer the mass and evaluate precision

Measure the unknown’s resonant frequency with the same method and find 1/f². Locate its corresponding mass on the calibration line, or use the fitted gradient and intercept. Choose calibration masses bracketing the likely unknown so the result is interpolated rather than extrapolated.

Because 1/f² depends on frequency squared, frequency uncertainty has an amplified effect. Repeat resonance judgements and compare the inferred mass with a balance measurement if permitted. A narrow peak helps identify frequency, but very large resonance amplitudes can change the spring behaviour or create practical hazards; use the supervised limits.

Work through the reasoning

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.

Example 1A calibration is

1/f^2=0.040M+0.0040

, with f in Hz and M in kg. An unknown resonates at 5.0 Hz. Find its added mass.

1/f^2=1/25=0.040

s².

M=(0.040-0.0040)/0.040=0.90

kg.

Answer0.90 kg; the intercept must be retained.

Example 2The gradient of 1/f² against mass is 0.040 s² kg⁻¹. Determine the spring constant in the ideal model.

1Gradient

=4\pi^2/k

k=4\pi^2/0.040=987

N m⁻¹.

AnswerAbout

9.9\times10^2

N m⁻¹.

Equation recap

\frac1{f^2}=\frac{4\pi^2}{k}(M+m_0)

Equation recap

Common mistakes to avoid

Forcing the calibration through zero despite fixed moving mass.

Define mass consistently and evaluate the intercept.

Using a different hanger only for the unknown.

Keep fixed moving parts represented by the calibration.

Key takeaways

Calibrate measured resonance frequency against known mass.
Inverse squared frequency gives a useful straight line.
Retain fixed-mass and damping effects in the evaluation.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 4 lessons