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The Pendulum

Source lecture(s): SC133 Lec 21

Intuition

A mass on a string, pulled aside and released, swings with a rhythm so steady it timed civilization's clocks for 300 years. For small swings it is simple harmonic motion with gravity playing the spring — and its most famous property is what it doesn't depend on: the mass and (for small angles) the amplitude.

The simple pendulum

Tangential Newton's law for a bob on a string of length \(L\) at angle \(\theta\):

\[mL\ddot\theta = -mg\sin\theta\]

Small-angle approximation \(\sin\theta \approx \theta\) (good to 1% below ~14°):

\[\ddot\theta = -\frac{g}{L}\,\theta \quad\Rightarrow\quad \boxed{\,\omega = \sqrt{\frac{g}{L}}, \qquad T = 2\pi\sqrt{\frac{L}{g}}\,}\]
  • Mass cancels — the same equivalence of inertial and gravitational mass behind Galileo's falling bodies.
  • Longer pendulum, slower swing: a 1 m pendulum ticks \(T \approx 2.0\) s (the historical "seconds pendulum" — one second per half-swing).
  • Measure \(T\) and \(L\), and you've measured \(g\): the classic first-year lab.

Beyond small angles, the period grows with amplitude (\(T \approx T_0[1 + \theta_0^2/16 + \dots]\)) — pendulum clocks needed small, constant swings.

The physical pendulum

Any rigid body swinging about a pivot: replace force balance with torque about the pivot,

\[I\ddot\theta = -mgd\sin\theta \quad\Rightarrow\quad T = 2\pi\sqrt{\frac{I}{mgd}}\]

where \(d\) is pivot-to-CM distance and \(I\) the moment of inertia about the pivot. Check: a rod pivoted at its end (\(I = \tfrac13 mL^2\), \(d = L/2\)) gives \(T = 2\pi\sqrt{2L/3g}\) — swings like a simple pendulum of length \(\tfrac23 L\).

Worked example: pendulum on the Moon

Your grandfather clock keeps perfect time on Earth. On the Moon (\(g_M = g/6\)), each period stretches by \(\sqrt6 \approx 2.45\): the clock runs slow by a factor 2.45. Pendulum clocks are gravimeters in disguise — expedition scientists once mapped \(g\) (and thus Earth's shape) by timing pendulums.

Energy view

\(U = mgL(1 - \cos\theta)\) from the lowest point; release from \(\theta_0\) gives bottom speed \(v = \sqrt{2gL(1 - \cos\theta_0)}\) — pure energy conservation, valid at any amplitude (unlike the small-angle period).

Common mistakes

  • Including the mass in the period. It cancels — a lead bob and a wooden bob of equal length keep identical time.
  • Using the small-angle formula for large swings — at \(\theta_0 = 90°\) the true period is ~18% longer.
  • Confusing \(L\) with the string length for a physical pendulum — you need \(I\) and \(d\), not just geometry's longest line.
  • Assuming tension does work — it's always ⊥ motion; only gravity trades energy.

Knowledge graph position

Prerequisites: SHM, Torque. Leads to: Damped oscillations; historically, to precision timekeeping and gravimetry.

Quiz

Q1 (computational). What length pendulum has a period of exactly 1 s on Earth?

Answer

\(L = g(T/2\pi)^2 = 9.8/(4\pi^2)\times1 \approx 0.248\,\text{m}\) — about 25 cm.

Q2 (conceptual). A pendulum clock is moved from sea level to a high mountain. Fast or slow?

Answer

Slow: \(g\) decreases with altitude, so \(T = 2\pi\sqrt{L/g}\) lengthens. (Correcting clocks like this is literally how \(g\)-variations were first surveyed.)

Q3 (multiple choice). A child stands up on a moving swing. The period: (a) increases (b) decreases (c) unchanged

Answer

(b). Standing raises the CM toward the pivot — shorter effective \(L\), shorter period. (Kneeling/standing rhythmically is also how you pump a swing.)