Free Pearson Edexcel International A Level Physics lesson
Analyse radiation counts and oscillation records
Random counts and damped motion need models that match the measurement.
What you’ll learn
- Plan and evaluate core practical 15.
- Use background correction and count uncertainty responsibly.
- Measure SHM, amplitude, period and damping for core practical 16.
Core practical 15: separate transmitted gamma counts from background
Measure gamma count rate through different known lead thicknesses with the source, absorber and detector held in a fixed geometry. Record background counts without the source contribution over a sufficiently long interval and convert all readings to rates before subtraction. The source must be handled only in an authorised supervised laboratory using local radiation procedures, appropriate shielding, handling tools and exposure controls.
Use a stack of measured lead sheets, keeping the source–detector distance and alignment unchanged as thickness changes. Gamma absorption is statistical: some photons pass through, so increasing thickness reduces the expected transmitted rate rather than giving a sharp range cutoff. Longer counting intervals reduce the relative influence of random count fluctuations, especially after substantial absorption.
An approximate narrow-beam attenuation model is I = I₀e^(−μx), so ln corrected rate against thickness can be linear with slope −μ. Scattered radiation, background changes and broad-beam geometry can cause departures. Do not take logs of zero or negative background-corrected rates; such a result indicates insufficiently resolved signal for that transformation, not negative physical intensity.
Rsource = Ntotal/ttotal − Nbackground/tbackground; ln(Rsource/Rref) = constant − μx
Symbols and units
- Ntotal and Nbackground are recorded counts
- ttotal and tbackground are measurement durations in s
- Rsource is net rate in s⁻¹
- μ is attenuation coefficient in inverse units of thickness x.
How the equation explains the idea
- Convert each count to a rate using its own duration before subtracting background. Subtracting raw counts from unequal times gives the wrong net rate.
- For an exponential attenuation model, ln(Rsource/Rref) = constant − μx, so the negative slope estimates μ. A zero or negative background-subtracted rate cannot be logarithmically transformed as though it were a positive signal.
Worked example
Different counting intervals
- A source-plus-background count is 720 in 120 s, giving 6.0 s⁻¹.
- Background is 300 in 300 s, giving 1.0 s⁻¹.
- Corrected source rate = 6.0 − 1.0 = 5.0 s⁻¹.
Subtract rates when counting intervals differ.
Keep this idea. Keep geometry fixed, subtract background rates and assess whether the remaining signal is resolved.
Random counts need long enough observations
Radioactive decay is random for an individual nucleus. Repeated counts over equal intervals therefore fluctuate even with unchanged apparatus and an approximately constant source activity. A longer observation generally contains more events and has smaller relative statistical fluctuation. A smooth expected decay curve describes the ensemble trend, not a promise that every measured point lies on it.
For independent counting with an appropriate Poisson model, a useful statistical standard uncertainty is approximately √N for N counts. This is different from the specification’s half-resolution or half-range convention for ordinary readings and should be identified as a counting model when used. Its relative value is 1/√N, so reducing it by half requires about four times as many counts.
Background subtraction combines uncertainties from both measured totals. If counts are taken for equal times, the approximate standard uncertainty of their difference is √(Ntotal + Nbackground), assuming independence; for unequal times propagate the rates with their own time factors. Do not replace a large background uncertainty by zero after subtraction. A decreasing source also requires a timing model if activity changes appreciably during each count.
Poisson model: σN ≈ √N; σR ≈ √N/t; independent rates: σnet² = Ntotal/ttotal² + Nbackground/tbackground²
Symbols and units
- N is a count and σN is its approximate Poisson standard uncertainty
- t is time in s
- R = N/t and σR are rates in s⁻¹.
How the equation explains the idea
- For Poisson counting, σN ≈ √N, hence σR ≈ √N/t. At fixed mean rate, quadrupling time quadruples expected counts and halves the fractional counting uncertainty.
- For independent source-plus-background and background measurements, variances of their rates add when rates are subtracted: σnet² = Ntotal/ttotal² + Nbackground/tbackground². Background subtraction does not subtract the uncertainty.
Conditions. Poisson source with mean detected rate 10 s⁻¹ and no background. The curve gives expected fractional standard uncertainty 100/√(rate × time); actual observed counts fluctuate.
Counting duration / s; Relative standard uncertainty / %.
At 10 s the expected count is 100 and relative uncertainty 10%; at 40 s it is 400 and relative uncertainty 5%. Four times the counting time halves this contribution, rather than reducing it fourfold.
Open the interactive lesson to view the plotted curves and their coordinate tables.
Worked example
Counting precision
- A count of 400 has approximate statistical uncertainty √400 = 20.
- Relative uncertainty is 20/400 = 5%.
- A count of 1600 has √1600 = 40, but relative uncertainty is 2.5%.
More counts increase absolute scatter while reducing relative scatter.
Keep this idea. Separate counting statistics from instrument resolution and include uncertainty in the background estimate.
Record simple harmonic motion and its departures
A mechanical SHM investigation records an oscillating mass or pendulum using a displacement sensor, video or a timing reference. Define equilibrium and measure displacement from it. For SHM, acceleration is proportional to displacement and opposite in direction, a = −ω²x. A sinusoidal displacement trace alone is suggestive; the restoring-force or acceleration relation explains the model.
For a mass–spring system, vary mass and measure many periods at each value to test T² proportional to mass. For a small-angle pendulum, vary pivot-to-centre length and test T² proportional to length. Use a fixed fiducial and count crossings in the same direction. Release without a push, keep amplitudes within the model range and secure the support and falling-mass zone.
Extract amplitude, period and phase from the trace, and observe how damping reduces amplitude over time. Repeating early and late timing windows tests whether the period remains approximately constant as amplitude falls. Strong damping or large pendulum angles can make the ideal relations inadequate. A forced-oscillation extension sweeps driving frequency carefully and allows transients to settle before comparing steady amplitudes near resonance.
a = −ω²x; x = A cos(ωt + φ); T = 2π√(m/k); small-angle pendulum T = 2π√(l/g)
Symbols and units
- x is displacement in m
- a is acceleration in m s⁻²
- ω is angular frequency in rad s⁻¹
- T is period in s
- m is mass in kg
- k is stiffness in N m⁻¹
- l is pendulum length in m.
How the equation explains the idea
- SHM requires acceleration opposite to and proportional to displacement, a = −ω²x. A plot of a against x has negative slope −ω².
- For a spring, ω² = k/m gives T = 2π√(m/k); for a small-angle pendulum, ω² = g/l gives T = 2π√(l/g). Large amplitude, damping or nonlinearity can change these ideal relations.
Conditions. Ideal SHM with angular frequency ω = 5 rad s⁻¹. Displacement is plotted in cm and converted to m in a = −ω²x.
Displacement / cm; Acceleration / m s⁻².
At +4 cm, acceleration is −1 m s⁻²; at −4 cm it is +1 m s⁻². The plotted gradient is −0.25 (m s⁻²) per cm; converting displacement to metres gives −25 s⁻² = −ω².
Open the interactive lesson to view the plotted curves and their coordinate tables.
Worked example
A spring from periods
- For mass 0.25 kg, 20 oscillations take 16.0 s, giving T = 0.800 s.
- k = 4π²m/T² = 4π² × 0.25/0.800².
- k = 15.4 N m⁻¹ approximately.
A set of masses and a T² graph tests the model better than one substitution.
Keep this idea. Measure complete cycles, define equilibrium and test the restoring relation and model limits.
Core practical 16: find an unknown mass from resonance
Attach known masses to the same spring or oscillating system and drive it at a controllable frequency with small approximately constant driving amplitude. For each known mass, sweep frequency through the response peak, wait for transients to settle and identify the frequency of greatest steady displacement amplitude. Use smaller frequency increments around the peak and repeat the sweep in both directions. The measured resonant frequency approximates the natural frequency only when damping is sufficiently light.
Calibrate using several known total attached masses over a range that brackets the unknown. For an ideal mass–spring system f = (1/2π)√(k/m), so 1/f² is proportional to mass. An effective spring or hanger mass gives a nonzero intercept; retain the same hanger and spring in the calibration and unknown measurements. Plot known mass against 1/f², or the reverse with the correct interpretation, and interpolate the unknown mass from its measured resonant frequency.
Keep the spring within its linear range, maintain the support and driving arrangement, and use a secure low-height mass attachment. Strong damping broadens the peak and may shift its maximum, increasing frequency uncertainty and weakening the simple model. Since 1/f² contains a square, frequency uncertainty makes roughly twice the fractional contribution to that transformed coordinate. This calibration is a mass determination from resonant frequencies, not merely a measurement of free-oscillation period.
Light damping: fᵣ ≈ (1/2π)√[k/(m + meffective)]; 1/fᵣ² = (4π²/k)(m + meffective)
Symbols and units
- fᵣ is resonance frequency in Hz
- k is spring stiffness in N m⁻¹
- m and meffective are attached and effective extra masses in kg.
How the equation explains the idea
- Under the light-damping approximation, fᵣ ≈ √[k/(m + meffective)]/(2π). Inverting and squaring gives 1/fᵣ² = (4π²/k)m + (4π²/k)meffective.
- The slope gives 4π²/k and the intercept divided by slope gives effective extra mass. Use calibration within its justified range; strong damping can shift the amplitude peak away from the natural frequency.
Worked example
Interpolate a mass from the calibration
- A fitted calibration is 1/fᵣ² = (2.0 s² kg⁻¹)m + 0.10 s².
- The unknown resonates at 2.0 Hz, so 1/fᵣ² = 0.25 s².
- m = (0.25 − 0.10)/2.0 = 0.075 kg.
Retaining the intercept avoids attributing the effective oscillator mass to the unknown.
Keep this idea. Calibrate resonance with known masses, retain the same oscillator and interpolate the unknown within the measured range.
Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.
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