Pendulums are systems that swing or twist because a restoring torque pulls them back toward equilibrium. This cheat sheet compares the simple pendulum, physical pendulum, and torsional pendulum so students can choose the correct model quickly. It is useful for solving oscillation problems, identifying small-angle conditions, and connecting torque, inertia, and period.
These ideas appear often in mechanics, rotational motion, and simple harmonic motion units.
The core idea is that a pendulum behaves like simple harmonic motion when its restoring torque is proportional to displacement. A simple pendulum uses , a physical pendulum uses , and a torsional pendulum uses . The period depends on inertia and restoring strength, not on mass alone.
Large amplitudes, damping, and driving forces require corrections beyond the ideal formulas.
Key Facts
- For a simple pendulum at small angles, the period is , where is the length from pivot to bob center.
- The small-angle approximation is when is measured in radians and is usually accurate for small amplitudes.
- The angular frequency of a simple pendulum is , so .
- For a physical pendulum, the period is , where is rotational inertia about the pivot and is the distance from pivot to center of mass.
- The parallel-axis theorem gives when converting rotational inertia from the center of mass to the pivot.
- For a torsional pendulum, the restoring torque is and the period is .
- Frequency and period are related by , and angular frequency is related by .
- Ideal pendulum period is independent of amplitude only when the oscillations are small enough for simple harmonic motion.
Vocabulary
- Simple pendulum
- A point mass on a light string or rod that swings under gravity with period for small angles.
- Physical pendulum
- An extended rigid body that swings about a pivot, with its mass distribution included through rotational inertia .
- Torsional pendulum
- A rotating system that twists back and forth because a wire or support produces restoring torque .
- Restoring torque
- A torque that acts opposite the displacement and tends to return the system to equilibrium.
- Rotational inertia
- A measure of resistance to angular acceleration, written , that depends on both mass and how far the mass is from the axis.
- Small-angle approximation
- The rule for in radians, which lets many pendulums be modeled as simple harmonic oscillators.
Common Mistakes to Avoid
- Using degrees inside is wrong because the approximation only works when is measured in radians.
- Using the bob mass in is wrong because the ideal simple pendulum period does not depend on mass.
- Using the total object length as for a physical pendulum is wrong because must be the distance from the pivot to the center of mass.
- Forgetting the parallel-axis theorem is wrong when the given is about the center of mass, because the physical pendulum formula needs about the pivot.
- Using for every pendulum is wrong because torsional pendulums use and physical pendulums use .
Practice Questions
- 1 A simple pendulum has length . Using , find its period .
- 2 A physical pendulum has about its pivot, mass , and center of mass distance . Find its small-angle period.
- 3 A torsional pendulum has rotational inertia and torsion constant . Calculate and .
- 4 Explain why increasing the mass of the bob does not change the ideal simple pendulum period, but changing the mass distribution of a physical pendulum can change its period.
Understanding Pendulums (Simple, Physical, Torsional)
For a hanging bob, gravity always points downward, but only part of gravity pulls the bob along its curved path. Near the lowest point, that part changes almost directly with the angle from vertical. This is why the motion has a nearly regular rhythm for gentle releases.
The approximation works best when the starting angle is small and measured in radians. A release from a wide angle takes slightly longer to complete each cycle.
Students often lose marks by using a large angle with an ideal small-angle model. They should check the diagram, identify the release angle, and state whether the approximation is reasonable.
A physical pendulum is any rigid object swinging about a pivot. Its behavior depends strongly on where its mass sits. Mass far from the pivot is harder to rotate because it gives a larger rotational inertia.
Moving the pivot along a meter stick, ruler, or metal plate can therefore change the period in a non-obvious way. The center of mass matters because gravity acts as though the object’s whole weight were concentrated there for torque calculations.
A pivot directly through the center of mass gives no gravitational turning effect, so the object will not swing back by itself. In calculations, students need the inertia about the actual pivot, not automatically the value listed for the center of mass.
A torsional pendulum turns back and forth around an axis rather than swinging through space. Examples include a disk hung from a thin wire, a balance wheel in a watch, and some sensitive instruments. Twisting the wire changes its internal structure slightly.
The material resists this twist and produces a torque toward its untwisted position. A stiffer wire gives a stronger restoring effect and a quicker oscillation.
A heavy disk with its mass near the rim turns more slowly than one with the same mass concentrated near the axis. This makes torsional systems useful for finding rotational inertia or comparing the stiffness of wires.
Real pendulums gradually lose energy. Air resistance, friction at a pivot, and internal rubbing in a wire convert organized motion into thermal energy. The amplitude then shrinks, though the period may remain close to its ideal value when damping is weak.
Strong damping can prevent repeated oscillations altogether. A clock pendulum needs a small regular input of energy to replace these losses. That input must be timed carefully.
A badly timed push can reduce the motion rather than support it. During lab work, students should time many complete cycles instead of one, start from a consistent position, avoid pushing the object sideways, and measure the correct length or pivot distance. These details often matter more than calculator precision.