Simple and Physical Pendulums
A physical pendulum is a rigid body oscillating about a fixed pivot. Gravity acts at the center of mass a distance d from the pivot, giving restoring torque τ=-mgd sinθ. For small amplitude, sinθ≈θ and I_p θ¨=-mgdθ, so ω=√(mgd/I_p) and T=2π√(I_p/(mgd)). A simple pendulum is the special point-mass case I_p=mℓ² and d=ℓ, giving T=2π√(ℓ/g). A torsion pendulum instead uses τ=-κθ and T=2π√(I/κ).
Topic sequence · 5 of 5
Learning objectives
Objective 1
Distinguish simple, physical, and torsion pendulum models and identify the correct restoring torque and inertia.
Objective 2
Derive the physical-pendulum period from Newton’s second law in rotational form and the small-angle approximation.
Objective 3
Use the parallel-axis theorem, functional dependence, and experimental linearization to solve or test pendulum models.
- Topic 5.4 rotational inertia and parallel-axis theorem.
- Topic 5.6 Newton’s second law in rotational form.
- Topics 7.1–7.2 SHM condition and period-frequency relations.
Definitions first
Representations you must read and create
Representation 1
Physical-pendulum diagram showing pivot, center of mass, distance d, weight mg, angle θ, and gravitational torque direction.
Representation 2
Comparison table: simple pendulum uses point-mass I=mℓ²; physical pendulum uses actual I_p; torsion pendulum uses κ rather than gravity.
Representation 3
T² versus an experimentally varied parameter (such as ℓ for a simple pendulum) to create a linear relationship and infer g.
Representation 4
Potential-energy view near stable equilibrium: gravitational potential has a local minimum and becomes approximately quadratic for small angular displacement.
Core relationships and calculus
Variables: m kg; g m/s²; d m; θ rad
Use: gravitational restoring torque of a physical pendulum
Variables: I_p kg·m²
Use: small-angle physical-pendulum equation
Variables: ω rad/s
Use: small-angle physical pendulum
Variables: T s
Use: small-angle physical pendulum
Variables: ℓ m
Use: point-mass simple-pendulum special case
Variables: I kg·m²
Use: parallel-axis theorem when I_cm is known
Variables: κ N·m/rad
Use: ideal torsion pendulum
Start from rotational dynamics about the fixed pivot: Στ=I_p α. Gravity produces τ=-mgd sinθ. For small angular amplitude, sinθ≈θ in radians, so I_p θ¨=-mgd θ and θ¨+(mgd/I_p)θ=0. Comparing with θ¨+ω²θ=0 gives ω²=mgd/I_p and T=2π/ω. The simple pendulum follows by setting I_p=mℓ² and d=ℓ. The torsion pendulum follows the same pattern from Iθ¨=-κθ.
If only I_cm is known for a physical pendulum, shift to the actual pivot before using the period formula: I_p=I_cm+md². For a uniform rod pivoted a distance a from its center, I_p=mL²/12+ma² and d=|a|, so T=2π√[(L²/12+a²)/(g|a|)]. This expression shows why moving the pivot changes both the restoring torque scale mgd and the inertia; there is no valid shortcut based only on 'longer distance means longer period.'
For gravitational pendulums, the SHM period formulas require small angular amplitude so sinθ≈θ. The physical-pendulum inertia is about the actual fixed pivot. The course also includes torsion pendulums with restoring torque proportional to angular displacement. Large-angle nonlinear pendulum period corrections and full damping theory are outside the required model unless a problem explicitly develops them.
Physical examples
Example 1
Uniform rod of length L pivoted at one end: I_p=(1/3)mL² and d=L/2, so T=2π√(2L/(3g)) for small angles.
Example 2
A simple pendulum is recovered from the physical formula by substituting I_p=mℓ² and d=ℓ; mass cancels.
Example 3
Moving the pivot of a rigid body changes both d and I_p, so period cannot be predicted from d alone.
Example 4
A torsion pendulum can oscillate in a horizontal plane because the restoring torque comes from twist, not gravity.
Example 5
For a physical pendulum, using I_cm instead of I_p generally underestimates the inertia and produces the wrong period.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Identify the oscillator before choosing a formula: point-mass simple pendulum, extended physical pendulum, or torsion pendulum.
Reasoning pattern 2
For a physical pendulum, locate the center of mass, find d, compute I_p about the pivot, then derive or use T=2π√(I_p/mgd).
Reasoning pattern 3
Use the parallel-axis theorem before substituting if the provided inertia is about the center of mass rather than the pivot.
Reasoning pattern 4
State sinθ≈θ in radians when deriving the small-angle equation.
Reasoning pattern 5
Use functional dependence to predict how period changes when the pivot, mass distribution, g, or torsion constant changes.
Reasoning pattern 6
For experimental design, time many cycles and linearize a relationship whose slope gives the target physical parameter.
Deeper calculus and limiting-case check
A physical pendulum can be assigned an equivalent simple-pendulum length ℓ_eq=I_p/(md), because T_phys=2π√(I_p/(mgd))=2π√(ℓ_eq/g). This does not mean the mass is physically concentrated at ℓ_eq; it is a period-equivalent mapping that is useful for comparisons and limiting-case checks.
Model-selection workflow: decide what the problem is before choosing an equation
For Simple and Physical Pendulums, the fastest reliable route is not equation hunting. Start by identifying the equilibrium point and restoring interaction, encode it with the phase/energy representation, choose the oscillator differential equation, and then test the small-displacement or ideal-oscillator assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat physical pendulum and simple pendulum as physical quantities first and symbols second.
Choose a representation
Build Physical-pendulum diagram showing pivot, center of mass, distance d, weight mg, angle θ, and gravitational torque direction.. If the prompt supplies Comparison table: simple pendulum uses point-mass I=mℓ²; physical pendulum uses actual I_p; torsion pendulum uses κ rather than gravity., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is τ_g = -mgd sinθ. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
For gravitational pendulums, the SHM period formulas require small angular amplitude so sinθ≈θ. The physical-pendulum inertia is about the actual fixed pivot. The course also includes torsion pendulums with restoring torque proportional to angular displacement. Large-angle nonlinear pendulum period corrections and full damping theory are outside the required model unless a problem explicitly develops them. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Physical-pendulum diagram showing pivot, center of mass, distance d, weight mg, angle θ, and gravitational torque direction. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | τ_g = -mgd sinθ | gravitational restoring torque of a physical pendulum. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | I_p θ¨ ≈ -mgd θ | small-angle physical-pendulum equation. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Start from rotational dynamics about the fixed pivot: Στ=I_p α. Gravity produces τ=-mgd sinθ. For small angular amplitude, sinθ≈θ in radians, so I_p θ¨=-mgd θ and θ¨+(mgd/I_p)θ=0. Comparing with θ¨+ω²θ=0 gives ω²=mgd/I_p and T=2π/ω. The simple pendulum follows by setting I_p=mℓ² and d=ℓ. The torsion pendulum follows the same pattern from Iθ¨=-κθ. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A uniform rod of length L and mass m is pivoted at one end and released from a small angle. About the end, I_p=(1/3)mL²; its center of mass lies d=L/2 from the pivot. The small-angle equation is I_p θ¨=-mgd θ, so θ¨=-[mg(L/2)/(mL²/3)]θ=-(3g/2L)θ. Therefore ω=√(3g/(2L)) and T=2π√(2L/(3g)). For L=1.00 m and g=9.80 m/s², T=1.64 s. Using the simple-pendulum formula with ℓ=L would be wrong because the rod’s mass is distributed rather than concentrated at its end. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 7.5
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship τ_g = -mgd sinθ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Physical-pendulum diagram showing pivot, center of mass, distance d, weight mg, angle θ, and gravitational torque direction. and Comparison table: simple pendulum uses point-mass I=mℓ²; physical pendulum uses actual I_p; torsion pendulum uses κ rather than gravity.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
For a simple pendulum, time many small-angle cycles at several lengths and plot T² versus ℓ; slope=4π²/g. For a physical pendulum such as a meter stick, choose a pivot away from the center of mass, measure d and T, calculate I_p from geometry/parallel-axis theorem, and compare measured T with 2π√(I_p/mgd). Alternatively, if g is known, rearrange the period equation to infer an unknown I_p and compare it with the geometric prediction. Report timing and length uncertainties.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using the simple-pendulum formula for an extended rigid body such as a rod or disk. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using I_cm in T_phys when the body actually rotates about a different pivot; use I about the pivot. | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Taking d as the object’s full length instead of the pivot-to-center-of-mass distance. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Applying sinθ≈θ without stating the small-angle condition or using degrees inside the approximation. | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming pendulum period is always amplitude-independent; the standard formulas are small-angle SHM results. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Confusing torsion constant κ (N·m/rad) with linear spring constant k (N/m). | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 5.4 supplies the rotational inertia and parallel-axis theorem required for physical pendulums. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 5.6 supplies Στ=Iα, which turns gravitational or torsional restoring torque into the SHM differential equation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 7.2 supplies the simple-pendulum period and square-root scaling; this topic derives the more general physical form. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write τ_g = -mgd sinθ and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 7.5 mastery ladder
Interactive pendulum model comparison
Compare small-angle simple, physical, and torsion pendulum periods. For the physical pendulum, I must be about the actual pivot and d is pivot-to-center-of-mass distance.
Experimental connection
For a simple pendulum, time many small-angle cycles at several lengths and plot T² versus ℓ; slope=4π²/g. For a physical pendulum such as a meter stick, choose a pivot away from the center of mass, measure d and T, calculate I_p from geometry/parallel-axis theorem, and compare measured T with 2π√(I_p/mgd). Alternatively, if g is known, rearrange the period equation to infer an unknown I_p and compare it with the geometric prediction. Report timing and length uncertainties.
Connections across Physics C
Connection 1
Topic 5.4 supplies the rotational inertia and parallel-axis theorem required for physical pendulums.
Connection 2
Topic 5.6 supplies Στ=Iα, which turns gravitational or torsional restoring torque into the SHM differential equation.
Connection 3
Topic 7.2 supplies the simple-pendulum period and square-root scaling; this topic derives the more general physical form.
Connection 4
Topic 7.4 connects pendulum turning points and equilibrium to kinetic/gravitational-potential energy exchange.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using the simple-pendulum formula for an extended rigid body such as a rod or disk.
Avoid this
Using I_cm in T_phys when the body actually rotates about a different pivot; use I about the pivot.
Avoid this
Taking d as the object’s full length instead of the pivot-to-center-of-mass distance.
Avoid this
Applying sinθ≈θ without stating the small-angle condition or using degrees inside the approximation.
Avoid this
Assuming pendulum period is always amplitude-independent; the standard formulas are small-angle SHM results.
Avoid this
Confusing torsion constant κ (N·m/rad) with linear spring constant k (N/m).
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What makes a pendulum physical rather than simple?
A physical pendulum has distributed mass and must use its actual rotational inertia about the pivot. A simple pendulum concentrates the mass at one point.
Why does the mass cancel for a simple pendulum?
Both the gravitational restoring torque and point-mass rotational inertia are proportional to m, so m cancels from the equation of motion.
Which moment of inertia belongs in the physical-pendulum formula?
The moment of inertia about the actual fixed pivot, not automatically I_cm.
What restores a torsion pendulum?
A twist-dependent torque τ=-κθ, rather than gravity, provides the linear restoring torque.