Beyond STEP
Chapter
11

Oscillations, Rotation and Moving Frames

Things that swing, things that spin, and things seen from somewhere that moves.

Download the whole chapterPDF, typeset for print

In this chapter

  1. 11.1 Resonance STEP II 1991 Worksheet
  2. 11.2 Normal Modes STEP III 2014 Worksheet
  3. 11.3 The Real Pendulum STEP III 1998 Worksheet
  4. 11.4 The Tautochrone STEP III 1994 Worksheet
  5. 11.5 Gravity in an Accelerating Train STEP II 2021 Worksheet
  6. 11.6 Backspin STEP III 1997 Worksheet
  7. 11.7 Simpson's Rule in a Rod STEP II 1989 Worksheet
  8. 11.8 Tilted Equilibria STEP III 2025 Worksheet
  9. 11.9 The Safety Parabola STEP II 2014 Worksheet
  10. 11.10 A Ball That Comes Back Towards You STEP II 2019 Worksheet
  11. 11.11 How Fielders Catch STEP II 1998 Worksheet
  12. 11.12 The Ant on the Rubber Rope STEP II 2018 Worksheet
  13. 11.13 The Snowplough Problem STEP III Specimen paper Worksheet
  14. 11.14 Parabola, Not Catenary STEP I 1991 Worksheet
  15. ✔ Looking back: what we have learned
Chapter 11 \(\cdot\) Introduction

Motion, Seen Clearly

The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.

— Attributed to Sidney Coleman

The previous chapter used conservation laws to understand collisions and gravity. This one turns to the other great themes of a first course in dynamics: things that oscillate, things that rotate, and the art of describing motion from the most useful point of view.

Oscillation is everywhere. Pendulums and springs are the textbook examples, but the same mathematics describes the vibrations of molecules, the swaying of buildings, the circuits in a radio and, in quantum mechanics, the behaviour of light itself. The reason is that almost anything near a stable equilibrium behaves like a spring: pull it a little way and the restoring force is proportional to the displacement. So the simple harmonic oscillator is the first approximation to almost every physical system, and much of physics consists of understanding what happens when the approximation is not quite right.

Why this matters

Engineers must know the natural frequencies of every structure they build, to keep them away from the frequencies of wind, traffic and earthquakes. Chemists identify molecules by their vibrations. Timekeeping, from Huygens’s pendulum clocks to quartz watches and atomic clocks, is the art of making a perfect oscillator. Rotation governs everything from the spin of a snooker ball to the stability of satellites. And the question of which frames of reference are “natural” led, through the equivalence of gravity and acceleration, to Einstein’s general relativity.

The chapter ends with six questions grouped as surprising trajectories: projectiles, ants, fielders and snowploughs whose behaviour is not what intuition predicts. They are short, elegant problems, and each has a proper mathematical idea at its centre: envelopes, monotonicity, feedback, divergence, linear differential equations and the difference between a parabola and a catenary.

What you will meet at university

A first-year dynamics course covers simple, damped and forced harmonic motion, resonance, and coupled oscillators and their normal modes (where linear algebra and mechanics meet). It treats rigid bodies, moments of inertia and rolling, and motion in accelerating and rotating frames, with the centrifugal and Coriolis forces. Later courses develop all of this into Lagrangian and Hamiltonian mechanics, where the choice of coordinates becomes a central theme, and into the theory of waves, which are oscillations spread out in space. Nonlinear oscillations, like the large-amplitude pendulum, lead to elliptic functions and to the dynamical systems of Chapter 8.

The questions that run through this chapter

Three big questions

  1. When is motion simple? When does a system oscillate with a single, fixed frequency, and what happens when it does not?

  2. What is the right point of view? Which coordinates, which variables and which frame of reference make a problem easy?

  3. Can intuition be trusted? Where does everyday intuition about motion go wrong, and how does mathematics correct it?

The chapter begins with oscillations: resonance (Section 11.1), normal modes, the real pendulum and the tautochrone. It then turns to frames and rigid bodies: an accelerating train, a spinning marble, a rod that behaves like three particles, and a solid that can rest at three different tilts. The last six sections are the surprising trajectories.

Ideas in this chapter

§ Topic Idea
11.1 Forced oscillations Resonance and beats
11.2 Coupled oscillators Normal modes are eigenvectors
11.3 The real pendulum Elliptic integrals; the AGM
11.4 The tautochrone Simple harmonic in arc length
11.5 Moving frames Fictitious forces; equivalence principle
11.6 Rolling and sliding A conserved angular momentum
11.7 Moments of inertia Dynamical equivalence; Simpson’s rule
11.8 Stability Normals through the centre of mass
11.9 Projectiles The envelope of trajectories
11.10 Projectiles Distance need not increase
11.11 Catching A feedback rule
11.12 Stretching ropes Fractions and the harmonic series
11.13 Snowploughs Time as a function of distance
11.14 Suspension bridges Parabola, not catenary

Two sections, Section 11.6 and Section 11.7, use rotational dynamics and moments of inertia, which were removed from the STEP specification in 2019. They are included because they are excellent questions; each gives the facts it needs.

How to read this chapter

Before you start any calculation, ask which variable the problem wants. Arc length for the tautochrone, \(x \pm y\) for coupled springs, the fraction of the rope for the ant, time as a function of distance for the snowplough: in each case, the right choice turns a difficult problem into a familiar one.