Physical Pendulum

A physical pendulum is any rigid body that is free to swing about a fixed pivot point under the influence of gravity. Unlike the idealised simple pendulum — which treats all the mass as concentrated at a single point on a massless string — a physical pendulum accounts for how mass is actually distributed throughout the object. A swinging door, a cricket bat pivoting about a batter's hands, or a metre rule pivoting about one end are all physical pendulums. The key insight is that the distribution of mass, not just the total mass, determines how quickly the object oscillates. The motion of a physical pendulum is governed by the same restoring-torque principle as the simple pendulum, but the period now depends on the moment of inertia of the body about the pivot. The moment of inertia captures how far each bit of mass sits from the axis of rotation — mass spread far from the pivot resists angular acceleration more than mass clustered near it. This is why a long, uniform rod swings more slowly than a compact disc of the same mass pivoting at its edge: the rod's mass is distributed further from the pivot on average. For small angular displacements (roughly less than 15°), the restoring torque is approximately proportional to the displacement angle, so the motion is simple harmonic. This small-angle approximation makes the mathematics tractable and leads to a clean formula for the period. Understanding physical pendulums is essential in engineering — designing seismometers, clock escapements, and even the human leg as a biomechanical pendulum all rely on these principles.

Key takeaways
  • The Restoring Torque
  • Applying Newton's Second Law for Rotation
  • Calculating the Moment of Inertia
  • Period of Oscillation
  • The Concept of Equivalent Length

The Restoring Torque

When a rigid body is displaced by a small angle θ from its equilibrium position, gravity acts downward through the body's centre of mass. If the centre of mass is a distance d from the pivot, gravity produces a restoring torque τ = −mgd sinθ. The negative sign means the torque always acts to return the body toward equilibrium. For small angles, sinθ ≈ θ (in radians), so τ ≈ −mgdθ — a linear restoring torque, which is the rotational analogue of Hooke's Law.

Applying Newton's Second Law for Rotation

Newton's second law for rotation states that the net torque equals the moment of inertia I times the angular acceleration α: τ = Iα. Substituting the restoring torque gives Iα = −mgdθ, or equivalently I(d²θ/dt²) = −mgdθ. Dividing both sides by I yields d²θ/dt² = −(mgd/I)θ. This is the equation of simple harmonic motion with angular frequency ω² = mgd/I. The moment of inertia I must be calculated about the pivot axis, not the centre of mass.

Calculating the Moment of Inertia

The moment of inertia about the pivot is rarely given directly. Instead, you often know I_cm — the moment of inertia about the centre of mass — from a table of standard results (e.g. I_cm = (1/12)mL² for a uniform rod about its centre). The parallel-axis theorem then gives I = I_cm + md², where d is the distance from the centre of mass to the pivot. This theorem is a crucial tool: it shifts the rotation axis from the centre of mass to any parallel axis.

Period of Oscillation

Since ω = 2π/T for any simple harmonic oscillator, rearranging gives T = 2π/ω = 2π√(I/mgd). Compare this with the simple pendulum T = 2π√(L/g): the physical pendulum replaces L with the effective length I/(md). If all the mass were at the centre of mass — a point mass on a massless rod of length d — you would get I = md², giving T = 2π√(d/g), exactly the simple pendulum formula. Real objects always have I > md², so they swing more slowly than a simple pendulum of length d.

The Concept of Equivalent Length

It is useful to define an equivalent (or reduced) pendulum length L_eq = I/(md). A simple pendulum of this length has exactly the same period as the physical pendulum. For a uniform rod of length L pivoting about one end: I = (1/3)mL², d = L/2, so L_eq = (1/3)mL² / (m·L/2) = (2/3)L. The rod therefore swings as though it were a simple pendulum of length two-thirds its actual length — a fact useful for quickly estimating periods without detailed calculation.

The Key Equation

T=2πImgdT = 2\pi\sqrt{\dfrac{I}{mgd}}
  • T — Period (s): Time for one complete oscillation of the pendulum.
  • I — Moment of inertia about pivot (kg m²): Measure of how the body's mass is distributed about the pivot axis; calculated using the parallel-axis theorem if needed.
  • m — Mass of rigid body (kg): Total mass of the swinging object.
  • g — Gravitational field strength (m s⁻²): Acceleration due to gravity; approximately 9.81 m s⁻² near Earth's surface.
  • d — Distance from pivot to centre of mass (m): The perpendicular distance from the fixed pivot point to the body's centre of mass.

Worked Example

Problem. A uniform thin rod of mass 0.50 kg and length 0.90 m is pivoted about one end and set oscillating with small amplitude. Calculate its period of oscillation. (Take g = 9.81 m s⁻².)

T≈1.55 sT \approx 1.55 \text{ s}

Answer. The period of oscillation is approximately 1.55 s.

Figure
Diagram
Centre of massdθPivot at topDisplaced rod (orange)Equilibrium rod (blue)Rigid rod, length LLTorque τ = −mgd sinθ restores the rod to equilibrium
Origins · 1 minChristiaan Huygens and how this was discovered
CH
Christiaan Huygens
1629–1695 · Dutch

Huygens derived the theory of the physical pendulum in his masterwork Horologium Oscillatorium (1673), one of the greatest scientific texts of the 17th century. He showed that any physical pendulum could be characterised by an equivalent simple pendulum length — what we now call the reduced length — and proved the reciprocity of pivot and centre of oscillation. He applied these results to design highly accurate pendulum clocks, revolutionising timekeeping. His work predated the formal concept of moment of inertia, which he effectively introduced in all but name. Huygens invented the pendulum clock in 1656 after being inspired by Galileo's earlier observations, reducing daily timekeeping errors from minutes to seconds — an accuracy improvement of roughly 60-fold overnight.

  • Simple Pendulum — The physical pendulum is a generalisation of the simple pendulum; understanding the idealised model first makes the transition to distributed-mass systems straightforward.
  • Simple Harmonic Motion — Physical pendulum oscillations under small-angle approximation are an example of SHM; the restoring-torque equation maps directly onto the SHM equation of motion.
  • Damped Oscillations — Real physical pendulums lose energy to air resistance and friction at the pivot, making damping an important correction to ideal periodic motion.
  • Wave Properties — Oscillating systems such as the physical pendulum are the fundamental source of wave motion; concepts of period and frequency transfer directly to wave physics.