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SHM systems: pendulums and springs

Two real oscillators carry the whole theory: a mass on a spring and a pendulum on a string. Each has a two-variable period formula, an energy see-saw between kinetic and potential, and a slow leak called damping.

Year 13AQA 3.6.1.3

Builds on Simple harmonic motion and Density and Hooke's law.

IN THIS TOPIC

  • Use T = 2π√(m/k) for a mass-spring system and T = 2π√(l/g) for a simple pendulum.
  • Describe how Ek, Ep and the total energy vary with displacement and with time.
  • Describe the effect of damping on an oscillation.

WHAT YOU PROBABLY THINK

Heavier pendulums swing slower.

The mass-spring clock

Hang a mass on a spring, displace it, and Hooke's law supplies exactly the restoring force SHM demands: F = −kx, so a = −(k/m)x, which is the defining equation with ω² = k/m. The period follows:

T = 2π√mkON YOUR DATA SHEET
The mass-spring clock: the period depends on the mass and the stiffness, and on nothing elsemmatters: m and kfour times the mass,twice the perioddoes not matter:amplitude, and g
FIG. 1A mass-spring oscillator. Its period depends on m and k alone: four times the mass, twice the period; amplitude and g never appear.

Read the formula like an ingredients list. More mass, more sluggish: quadruple m and T doubles. A stiffer spring, quicker: k sits underneath. And two famous absences: the amplitude, because SHM is isochronous, and g, because gravity only shifts the equilibrium point the mass bounces about, without touching the bounce itself. The same clock keeps the same time in orbit.

The pendulum clock

A pendulum's restoring force is the component of gravity along its arc, and for small angles that component is proportional to displacement, giving approximate SHM with

T = 2π√lgON YOUR DATA SHEET
The pendulum clock: the period depends on the length and on g, not on the mass or the (small) swingsmall swings only: the SHM is an approximationmatters: l and gfour times the length,twice the perioddoes not matter:the mass on the end
FIG. 2A simple pendulum. The period depends on l and g; the bob's mass cancels, and the SHM only holds for small swings.

The mass is absent: the restoring force and the inertia both scale with m, so it cancels, exactly as it did for objects in free fall. What does appear is g, which is why a pendulum makes a portable gravity meter. AQA flags the small-angle approximation explicitly: the formula, and the SHM itself, are approximations that hold for small swings. Other oscillators, liquid in a U-tube is the spec's example, can appear in questions, but with all the necessary information provided.

The energy see-saw

The energy see-saw: potential grows as the square of displacement, kinetic fills the gap, the total never movesxtotal: constantEpEkall kinetic at the middle, all potential at the ends
FIG. 3Energy against displacement: potential grows as x², kinetic fills the remainder, and the total sits constant across the swing.

The oscillation is a continuous trade. Potential energy is maximal at the extremes and zero at the centre, growing as the square of displacement; kinetic energy is its mirror, all at the centre, none at the ends; the total stays constant. Against time, each energy oscillates at twice the motion's frequency, since both extremes of a cycle look identical to the energy books. Sketching either picture, energy against displacement or against time, is a standard question.

Damping

Damping drains the energy: the oscillation keeps its rhythm while the amplitude decays awaytthe envelope: exponential decayenergy leaves each cycle; the period barely changes
FIG. 4A damped oscillation: the amplitude decays inside an exponential envelope while the period barely changes.

Real oscillators leak. Damping is the loss of energy to resistive forces, air resistance, internal friction, and its signature is an amplitude that decays, exponentially for the common case, while the period barely changes. The energy leaves as internal energy in the surroundings; the rhythm survives, quieter each cycle, until it fades entirely.

THE EXAM BIT

  • Match the formula to the system, and note neither contains amplitude: T = 2π√(m/k) for springs, T = 2π√(l/g) for pendulums.
  • The pendulum's period is independent of mass; the spring's is independent of g. Each absence is its own question.
  • Both formulas hide a square root: quadrupling m or l doubles T. Halving T needs a quarter of the length, not half.
  • Energy against time oscillates at twice the frequency of the motion; the total is a flat line. Both features are marked in sketches.
  • State the small-angle condition when using the pendulum formula; for large swings the motion is not simple harmonic and the formula overreaches.

CHECK YOURSELF

What length of simple pendulum has a period of 2.0 s? Take g = 9.81 m s−2. Would the answer change on the Moon?

Show a hint

Rearrange for l, and remember what sits inside the square root.

Show the answer

Rearranging: l = g(T/2π)2 = 9.81 × (2.0/2π)2 = 0.99 m, the metre-long “seconds pendulum” of old clocks.

On the Moon, yes: g is about a sixth, so the same 2.0 s period needs l = 1.62 × (0.318)2 ≈ 0.16 m. The pendulum reads its local gravity.

The bob's mass, note, was never asked for and never needed.

Springs count m and k; pendulums count l and g.

The energy see-saws; the total holds still.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.