Physical & Torsional Pendulums

Real swinging objects and twisting wires follow the same rhythm with I in charge.

11 minUpdated 21 October 2026physical pendulumcompound pendulumtorsional pendulumtorsion constant

Lesson at a glanceShow the key points

Physical pendulum: a rigid body swinging about a pivot a distance l from its centre of mass; for small angles T = 2π√(I/(mgl)) (12.24), with I about the pivot.
Its equivalent simple pendulum length is I/(ml).
A uniform rod pivoted at one end has T = 2π√(2L/3g).
Torsional pendulum: a twisted wire exerts torque −kθ; T = 2π√(I/k) (12.25), exact for any twist the wire tolerates.
Torsion pendulums measure tiny torques (the Cavendish balance) and moments of inertia.

Real swinging things are not point masses on threads. A ring on a nail, a swinging leg, a door closer: their rhythms follow the same rule once we let the moment of inertia take charge.

The physical pendulum

Any rigid body hung from a fixed pivot O is a physical (or compound) pendulum. It hangs at rest with its centre of mass C directly below O. Let OC = l. Turned through θ, the support force at O has no torque, and the weight has torque

\Gamma = -mgl\sin\theta \approx -mgl\,\theta \quad\Rightarrow\quad \alpha = -\frac{mgl}{I}\theta

Physical pendulum (small swings)(12.24)

T = 2\pi\sqrt{\frac{I}{mgl}}

I is the moment of inertia about the pivot, l the pivot-to-centre-of-mass distance

Comparing with 2π√(L/g), it swings like a simple pendulum of length $L = I/(ml)$ , the equivalent length. For a point bob, $I = ml^2$ and L = l, as it should.

Interactive

A swinging rod

A uniform 1 m rod hangs from a pivot a distance d from its centre. T = 2π√(I/mgd), with I about the pivot. Pivot at the end and you get 2π√(2L/3g). Move the pivot toward the centre and the period first falls, reaches a minimum at d = L/√12, then grows without limit.

I = \tfrac{1}{12}mL^2 + md^2

0.333m (×m)

T

1.639s

equivalent simple pendulum0.667m

Pivot distance from the centre

d

0.50m

Worked example

A ring on a nail

A thin uniform ring of radius 20 cm hangs on a nail through its rim and is set swinging gently in its own plane. Find the period.

The torsional pendulum

Hang a body (a disc, say) from a thin wire and twist it through θ about the wire. The wire twists too, and its elasticity produces a restoring torque proportional to the twist, $\Gamma = -k\theta$ , where k is the wire’s torsional constant. This is exactly angular SHM:

Torsional pendulum(12.25)

T = 2\pi\sqrt{\frac Ik}

I about the wire; valid for any twist the wire tolerates, not just small ones

Interactive

A twisting disc

A disc hangs from a wire. Twist it and let go: the wire's restoring torque is −kθ, so the disc twists back and forth in angular SHM with T = 2π√(I/k). (The animation is slowed for short periods.)

I = \tfrac12MR^2

5.40× 10⁻⁴ kg·m²

T = 2\pi\sqrt{I/k}

0.501s

Disc mass

M

0.30kg

Disc radius

R

6.0cm

Torsion constant

k

0.085N·m/rad

Worked example

Finding a wire's torsional constant

A uniform disc of mass 300 g and radius 6.0 cm is fixed at its centre to a vertical wire. Twisted and released, it oscillates with a period of 0.50 s. Find the torsional constant.

What to remember

1Physical pendulum: $T = 2\pi\sqrt{I/mgl}$ , with I about the pivot; equivalent length I/ml.
2A uniform rod pivoted at one end: T = 2π√(2L/3g).
3Torsional pendulum: $\Gamma = -k\theta$ , $T = 2\pi\sqrt{I/k}$ .

Check yourself

A metre stick swinging from one end versus a simple pendulum 1 m long: the stick's period is…

Doubling the radius of the disc on a torsion wire (same mass) multiplies the period by…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchHooke's law, springs, pendulums, SHMWalter Lewin · MIT 8.01x Lecture 10 ReadPendulumsOpenStax University Physics §15.4 ReadPhysical pendulumHyperPhysics ReadTorsion spring and torsion pendulumWikipedia

PreviousThe Simple Pendulum Next Combining Two Oscillations

Physical & Torsional Pendulums

Real swinging objects and twisting wires follow the same rhythm with I in charge.

11 minUpdated 21 October 2026physical pendulumcompound pendulumtorsional pendulumtorsion constant

Lesson at a glanceShow the key points

Physical pendulum: a rigid body swinging about a pivot a distance l from its centre of mass; for small angles T = 2π√(I/(mgl)) (12.24), with I about the pivot.
Its equivalent simple pendulum length is I/(ml).
A uniform rod pivoted at one end has T = 2π√(2L/3g).
Torsional pendulum: a twisted wire exerts torque −kθ; T = 2π√(I/k) (12.25), exact for any twist the wire tolerates.
Torsion pendulums measure tiny torques (the Cavendish balance) and moments of inertia.

The physical pendulum

\Gamma = -mgl\sin\theta \approx -mgl\,\theta \quad\Rightarrow\quad \alpha = -\frac{mgl}{I}\theta

Physical pendulum (small swings)(12.24)

T = 2\pi\sqrt{\frac{I}{mgl}}

I is the moment of inertia about the pivot, l the pivot-to-centre-of-mass distance

Comparing with 2π√(L/g), it swings like a simple pendulum of length

L = I/(ml)

, the equivalent length. For a point bob,

I = ml^2

and L = l, as it should.

Interactive

A swinging rod

I = \tfrac{1}{12}mL^2 + md^2

0.333m (×m)

T

1.639s

equivalent simple pendulum0.667m

Pivot distance from the centre

d

0.50m

Worked example

A ring on a nail

A thin uniform ring of radius 20 cm hangs on a nail through its rim and is set swinging gently in its own plane. Find the period.

The torsional pendulum

Hang a body (a disc, say) from a thin wire and twist it through θ about the wire. The wire twists too, and its elasticity produces a restoring torque proportional to the twist,

\Gamma = -k\theta

, where k is the wire’s torsional constant. This is exactly angular SHM:

Torsional pendulum(12.25)

T = 2\pi\sqrt{\frac Ik}

I about the wire; valid for any twist the wire tolerates, not just small ones

Interactive

A twisting disc

I = \tfrac12MR^2

5.40× 10⁻⁴ kg·m²

T = 2\pi\sqrt{I/k}

0.501s

Disc mass

M

0.30kg

Disc radius

R

6.0cm

Torsion constant

k

0.085N·m/rad

Worked example

Finding a wire's torsional constant

A uniform disc of mass 300 g and radius 6.0 cm is fixed at its centre to a vertical wire. Twisted and released, it oscillates with a period of 0.50 s. Find the torsional constant.

What to remember

1Physical pendulum: $T = 2\pi\sqrt{I/mgl}$ , with I about the pivot; equivalent length I/ml.
2A uniform rod pivoted at one end: T = 2π√(2L/3g).
3Torsional pendulum: $\Gamma = -k\theta$ , $T = 2\pi\sqrt{I/k}$ .

Check yourself

A metre stick swinging from one end versus a simple pendulum 1 m long: the stick's period is…

Doubling the radius of the disc on a torsion wire (same mass) multiplies the period by…

The physical pendulum#

The torsional pendulum#

Check yourself

Keep exploring

The physical pendulum#

The torsional pendulum#

Check yourself

Keep exploring

The physical pendulum

The torsional pendulum

The physical pendulum

The torsional pendulum