Episode 303: Mass-spring systems

This episode and the next focus on practical simple harmonic motion (SHM) systems.

Summary

  • Discussion: Hooke’s law leads to SHM (15 minutes)
  • Student experiment: Testing the relationship T = 2p√ (m/k) (30 minutes)
  • Student activity: Using a computer model (20 minutes)
  • Discussion: Modelling in physics (10 minutes)
  • Worked example: Applying the relationship T = 2p√ (m/k) (10 minutes)
  • Student questions: Calculations involving T = 2p√ (m/k) (30 minutes)

Discussion: Hooke’s law leads to SHM
If your specification requires it, here is where you can derive the expression for the period of a mass-spring system.

We have w = 2pf = √ (F/mx) as the requirement for SHM, and F = kx from Hooke’s law.

It is easiest to deal with a horizontal mass-spring system first (because you can ignore gravity).

Oscillating spring: force and displacement

Student experiment: Testing the relationship T = 2p√ (m/k).
Students can test the relationship T = 2p√ (m/k) for a mass-spring system. (Note that this expression is independent of g.)

Students may find that there is a systematic error, caused by the finite mass of the spring. Try modifying the simple theory to take into account the mass of the spring mS:

T = 2p [(m + mS)/k] 1/2

Episode 303-1: Loaded spring oscillator (Word, 59 KB)

Student activity: Using a computer model
They can also use a computer model.

Episode 303-2: Oscillating freely (Word, 26 KB)

and the model

Episode 303-3: Modelling springs and masses (Word, 24 KB)

Discussion: Modelling in physics
The study of SHM may well be the first occasion that students meet detailed mathematical modelling. It may be worth spelling out for them what is happening. There is a physical behaviour we want to understand. First we simplify the actual situation to an idealised physical model by making assumptions, e.g. no ‘friction’, pendulum strings or springs that have no mass, etc. Then we make a mathematical model to represent the physical model. The mathematical model is then analysed (‘solved’) and has to be interpreted in terms of the physical model. Experiments try to mirror the physical model but they cannot do this exactly (e.g. make a pendulum string as light as possible while still being strong enough to support the ‘bob’). So care is needed when comparing the theory with experimental measurements.

Worked example: Applying the relationship T = 2p√ (m/k)
A vibrating atom in a solid can be modelled as a mass m between two tensioned springs, the springs representing the interatomic forces.

For typical interatomic forces k = 60 N m-1

Mass of an atom (Na in NaCl) ~ 3.8 ´ 10-26 kg

Estimate the natural vibration frequency of atoms.

f = 1/T = (1/2p) [2k/m] 1/2

f ~ 9 ´ 1012 Hz, which is in the IR region of the electromagnetic spectrum. We will return to atomic vibrations when discussing resonance.

Student questions: Calculations involving T = 2p√ (m/k).
These questions reinforce basic ideas about SHM.

Episode 303-4: Harmonic oscillators (Word, 35 KB)

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Episode 303: Mass-spring systems (Word, 134 KB)