Pendulums, springs, energy and resonance

Key ideas

Two systems in the course oscillate in SHM, each with its own period:
- Simple pendulum (small swings): $T = 2\pi\sqrt{\dfrac{l}{g}}$ $T = 2 π \frac{l}{g}$
  - $l$ — the length of the pendulum (m)
  - the period does not depend on the mass or the amplitude (for small swings)
- Mass on a spring: $T = 2\pi\sqrt{\dfrac{m}{k}}$ $T = 2 π \frac{m}{k}$
  - $m$ — the mass (kg)
  - $k$ — the spring constant (N m⁻¹)
Energy in SHM continuously swaps between kinetic and potential, but the total stays constant (with no friction):
- all kinetic at the centre (maximum speed)
- all potential at the ends (momentarily at rest)
- total energy $\propto A^2$ — doubling the amplitude quadruples the energy.

Damping, driving and resonance

Damping — a resistive force (friction, air resistance) removes energy, so the amplitude decays over time. The period is barely changed for light damping.
Free oscillation happens at the system's own natural frequency $f_0$ .
Driven (forced) oscillation — an external periodic push. When the driving frequency matches the natural frequency, the amplitude grows very large: this is resonance.
More damping gives a lower, broader resonance peak. Engineers add damping (car suspension, tall buildings) precisely to limit resonance.

An amplitude-versus-driving-frequency graph showing a tall narrow resonance peak for light damping and a lower, broader peak for heavy damping, both peaking near the natural frequency.

Worked ExamplePeriod of a pendulum

Find the length of a simple pendulum whose period is exactly $2.0$ s (a "seconds pendulum"). Take $g = 9.81$ m s⁻².

Step 1 — Start from the pendulum period

$T = 2\pi\sqrt{\frac{l}{g}}$

Step 2 — Rearrange for the length $l$

$l = \frac{gT^2}{4\pi^2}$

Step 3 — Substitute (keep unrounded values)

$l = \frac{(9.81)(2.0)^2}{4\pi^2} = \frac{39.24}{39.48} = 0.994 \text{ m}$

Tips

Match the period formula to the system: $\sqrt{l/g}$ for a pendulum, $\sqrt{m/k}$ for a spring. Notice the mass helps a pendulum not at all but slows a spring — a favourite Excellence discriminator.
For resonance questions, always name the two frequencies you are comparing (driving vs natural) and state that they are equal at resonance; a vague "the frequencies match" often misses the mark.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A $0.25$ kg mass on a spring of constant $100$ N m⁻¹ oscillates in SHM.

Show that its period is about $0.31$ s.

Merit

A pendulum has length $1.5$ m. Take $g = 9.81$ m s⁻².

Calculate its period, and state what would happen to the period if a heavier bob were used.

Excellence

Soldiers are told to break step when crossing a footbridge.

Explain, using natural frequency, resonance and damping, why marching in step could be dangerous.

Pendulums, springs, energy and resonance

Covers AS91524

Key ideas

Two systems in the course oscillate in SHM, each with its own period:
- Simple pendulum (small swings): $T = 2\pi\sqrt{\dfrac{l}{g}}$ $T = 2 π \frac{l}{g}$
  - $l$ — the length of the pendulum (m)
  - the period does not depend on the mass or the amplitude (for small swings)
- Mass on a spring: $T = 2\pi\sqrt{\dfrac{m}{k}}$ $T = 2 π \frac{m}{k}$
  - $m$ — the mass (kg)
  - $k$ — the spring constant (N m⁻¹)
Energy in SHM continuously swaps between kinetic and potential, but the total stays constant (with no friction):
- all kinetic at the centre (maximum speed)
- all potential at the ends (momentarily at rest)
- total energy $\propto A^2$ — doubling the amplitude quadruples the energy.

Damping, driving and resonance

Damping — a resistive force (friction, air resistance) removes energy, so the amplitude decays over time. The period is barely changed for light damping.
Free oscillation happens at the system's own natural frequency $f_0$ .
Driven (forced) oscillation — an external periodic push. When the driving frequency matches the natural frequency, the amplitude grows very large: this is resonance.
More damping gives a lower, broader resonance peak. Engineers add damping (car suspension, tall buildings) precisely to limit resonance.

An amplitude-versus-driving-frequency graph showing a tall narrow resonance peak for light damping and a lower, broader peak for heavy damping, both peaking near the natural frequency.

Worked ExamplePeriod of a pendulum

Find the length of a simple pendulum whose period is exactly $2.0$ s (a "seconds pendulum"). Take $g = 9.81$ m s⁻².

Step 1 — Start from the pendulum period

$T = 2\pi\sqrt{\frac{l}{g}}$

Step 2 — Rearrange for the length $l$

$l = \frac{gT^2}{4\pi^2}$

Step 3 — Substitute (keep unrounded values)

$l = \frac{(9.81)(2.0)^2}{4\pi^2} = \frac{39.24}{39.48} = 0.994 \text{ m}$

Tips

Match the period formula to the system: $\sqrt{l/g}$ for a pendulum, $\sqrt{m/k}$ for a spring. Notice the mass helps a pendulum not at all but slows a spring — a favourite Excellence discriminator.
For resonance questions, always name the two frequencies you are comparing (driving vs natural) and state that they are equal at resonance; a vague "the frequencies match" often misses the mark.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A $0.25$ kg mass on a spring of constant $100$ N m⁻¹ oscillates in SHM.

Show that its period is about $0.31$ s.

Merit

A pendulum has length $1.5$ m. Take $g = 9.81$ m s⁻².

Calculate its period, and state what would happen to the period if a heavier bob were used.

Excellence

Soldiers are told to break step when crossing a footbridge.

Explain, using natural frequency, resonance and damping, why marching in step could be dangerous.

Pendulums, springs, energy and resonance

Key ideas

Damping, driving and resonance

Step 1 — Start from the pendulum period

Step 2 — Rearrange for the length lll

Step 3 — Substitute (keep unrounded values)

Tips

Test yourself

Practice by grade

Pendulums, springs, energy and resonance

Key ideas

Damping, driving and resonance

Step 1 — Start from the pendulum period

Step 2 — Rearrange for the length lll

Step 3 — Substitute (keep unrounded values)

Tips

Test yourself

Practice by grade

Step 2 — Rearrange for the length $l$

Step 2 — Rearrange for the length $l$