Investigating damping of a spring

Practical method

Aim

Investigate the damping of a mass–spring oscillator and show that the amplitude decays exponentially.

Variables

Independent variable: time (and the area of the damping card)
Dependent variable: amplitude
Control variables: the same mass and spring; the same initial displacement.

Apparatus

• Spring, mass and clamp stand
• Card discs of different areas to attach to the mass
• Metre rule beside the mass
• Stopwatch or video camera

Method

1. Attach a card disc to the mass to increase air resistance.
2. Displace the mass 10.0 cm and release it; video the motion against the metre rule.
3. From the video, read the amplitude every 5 s.
4. Plot ln A against t: a straight line shows exponential decay, A = A₀\(e^{-bt}\).
5. Repeat with larger cards and compare the damping constants.

Sample results

Card disc of diameter 10 cm, A₀ = 10.0 cm.

Analysis

ln A falls linearly with gradient −0.080 s⁻¹, so the amplitude decays exponentially with damping constant 0.080 s⁻¹. The amplitude halves every \(\dfrac{\ln 2}{0.080}\) = 8.7 s. Larger cards give steeper lines (heavier damping); the period hardly changes for light damping.

Uncertainty

Reading the amplitude from video to ±2 mm is 1% at 10 cm but 10% at 2 cm; later readings are least reliable.

Errors and improvements

Risk assessment

Common mistakes

• Plotting A against t and calling the curve linear.
• Measuring peak-to-peak and calling it amplitude.
• Using log₁₀ and quoting the gradient as b.

Worked analysis

Gradient of ln A against t = \(\dfrac{0.703 - 2.303}{20}\) = −0.080 s⁻¹. Time for the amplitude to halve = \(\dfrac{\ln 2}{0.080}\) = 8.7 s.

Common misconception

Light damping reduces the amplitude over time but barely changes the period; critical damping returns the system to equilibrium fastest without oscillating.