Required Practicals/AQA/Practical 7
7A23.6.1.2

Simple harmonic motion: mass-spring and pendulum

Investigate how the period of a mass-spring system and a simple pendulum depends on mass (spring) and length (pendulum).

Apparatus

  • Helical spring, selection of masses and hanger
  • Simple pendulum (string and bob), protractor
  • Stopwatch or light-gate and datalogger
  • Metre rule and stand with clamp
  • Balance to measure mass

Safety

  • Do not overload the spring beyond its elastic limit; permanent deformation would change k.
  • Keep the pendulum amplitude small (<10 degrees) for the SHM approximation to apply.

Method

  1. Mass-spring: hang the spring vertically and add mass m. Displace the mass ~1 cm downwards and release. Time 20 complete oscillations; divide by 20 to find T.
  2. Repeat for at least six different masses, recording T for each.
  3. Plot $T^2$ against m: straight line through origin, gradient $= 4\pi^2/k$.
  4. Pendulum: set up a pendulum of length L (pivot to centre of bob). Displace by <10 degrees and time 20 oscillations.
  5. Repeat for at least six lengths. Plot $T^2$ against L: gradient $= 4\pi^2/g$.

Key Variables

Independent
Mass m (spring) or length L (pendulum)
Dependent
Period T
Controlled
Spring constant k (spring method); g and amplitude <10 degrees (pendulum)

Analysis and Results

  • Mass-spring: $T = 2\pi\sqrt{m/k}$, so $T^2 = (4\pi^2/k)\,m$. Gradient of $T^2$ vs m gives $k = 4\pi^2/\text{gradient}$.
  • Simple pendulum: $T = 2\pi\sqrt{L/g}$, so $T^2 = (4\pi^2/g)\,L$. Gradient of $T^2$ vs L gives $g = 4\pi^2/\text{gradient}$.
  • Both graphs should be straight lines through the origin, confirming the proportionality.

Common Errors

  • Timing too few oscillations (e.g. 5), increasing the percentage uncertainty in T.
  • Measuring the pendulum length to the top of the bob rather than its centre of mass.
  • Using large amplitudes, which cause the period to increase and deviate from the SHM formula.
  • Not including the hanger mass in m for the spring experiment.

Exam-style questions on this practical. Click Show mark scheme to reveal the answer after attempting each question.

Q13 marks
A mass of 0.30 kg on a spring oscillates with period 0.85 s. Calculate the spring constant k.
Q23 marks
A student plots $T^2$ against L for a simple pendulum and obtains a gradient of 4.05 s$^2$ m$^{-1}$. Calculate the value of g.
Q32 marks
Explain why the amplitude of the pendulum must be kept small.