Unit 7 · Topic 09

Pendulums

Pendulum periods are approximations for small oscillations about equilibrium.

Unit 7PendulumsOpen navigation

Concept 01

Ideal point mass

Essential The minimum you should retain

For length L and small angle, T≈2πsqrt(L/g).

UnderstandInterpret and connect

The period is independent of mass and approximately independent of amplitude in that limit.

DeepenFormulation and conditions

The approximation uses sinθ≈θ.

ExploreConnections for further study

At larger amplitude the real period exceeds the simple approximation.

Worked example

Simple-pendulum period

An ideal L=1.00 m pendulum swings through a small angle; g=9.8 m/s².

Given
  • L=1.00 m
  • g=9.8 m/s²
  • small angle
Target

Find T.

  1. Model

    T≈2πsqrt(L/g).

  2. Substitution

    T≈2πsqrt(1/9.8).

  3. Calculation

    T≈2.007 s.

  4. Validity

    The approximation requires small oscillations.

Conclusion

T≈2.01 s.

Mathematical relation

Pendulums

Tsimple≈2πLg,Tfísico≈2πIpMgd T_{simple}\approx2\pi\sqrt{L/g},\quad T_{physical}\approx2\pi\sqrt{I_p/(Mgd)}
Represents

Approximate periods of simple and physical pendulums.

Physical interpretation

The simple formula is not exact at arbitrary amplitude.

DeepenVariables, conditions, and checks

Variables

I_p
moment of inertia about the pivot; usual unit: kg·m²
d
pivot-to-CM distance; usual unit: m

Conditions of application

  • Small oscillations; sinθ≈θ.

Dimensional check

s.

Two pendulums show pivot, center of mass, and small-oscillation parameters.Lrigid bodydpivotmpivotCMsmall θsmall θ

The simple pendulum uses L; the physical pendulum requires I_p and distance d from pivot to center of mass.

Concept 02

A stated limit

Essential The minimum you should retain

The formula is not exact for arbitrary amplitudes.

UnderstandInterpret and connect

Small angles keep the angular restoring force nearly linear.

DeepenFormulation and conditions

Amplitude corrections appear when nonlinear terms in sinθ are retained.

ExploreConnections for further study

Stating amplitude or regime prevents extrapolation.

Concept 03

Rigid distribution

Essential The minimum you should retain

For small oscillations, T≈2πsqrt(I_p/(Mgd)).

UnderstandInterpret and connect

I_p is measured about the pivot and d joins pivot to center of mass.

DeepenFormulation and conditions

The body cannot automatically be replaced by a point mass at its CM.

ExploreConnections for further study

Moving the pivot changes both I_p and d.

Worked example

Rod as a physical pendulum

A uniform L=1.00 m rod pivots at one end; I_p=(1/3)ML² and d=L/2.

Given
  • L=1.00 m
  • I_p=(1/3)ML²
  • d=L/2
  • g=9.8 m/s²
Target

Find T.

  1. Model

    T=2πsqrt(I_p/(Mgd)).

  2. Substitution

    Insert I_p and d.

  3. Reduction

    T=2πsqrt(2L/(3g)).

  4. Result

    T≈1.64 s.

Conclusion

T≈1.64 s for small oscillations.

Concept 04

Simple as a limit

Essential The minimum you should retain

The simple pendulum concentrates all mass a distance L from the pivot.

UnderstandInterpret and connect

The physical pendulum retains the body's mass distribution.

DeepenFormulation and conditions

Both models share the small-oscillation limit.

ExploreConnections for further study

Choosing the correct model is part of the reasoning.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using T=2πsqrt(L/g) exactly at every amplitude.

It is a small-angle approximation.

Automatically treating a physical pendulum as a point mass.

Use the full body's I_p and distance d.