KEVOS® MAKEOS · Mechanics
Pendulum Period: Simple, Compound, Conical and Torsional
What is the period of this pendulum, and how much does a large swing change it?
Period and frequency of a simple pendulum, a compound (physical) pendulum from its radius of gyration, a conical pendulum, and a torsional pendulum made of a disc on a slender rod, with a large-amplitude correction for swinging pendulums.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Pendulum | choice | Choose the pendulum type. |
| Length (cord, or rod for torsional) | m | Cord length from pivot to the bob centre, or the free length of the torsion rod. |
| Swing amplitude (simple, compound) | ° | Amplitude of swing either side of vertical; small swings need no correction. |
| Radius of gyration about the pivot (compound) | m | Radius of gyration of the body about the pivot axis. |
| Pivot to centre of gravity (compound) | m | Distance from the pivot to the body’s centre of gravity. |
| Cord angle from vertical (conical) | ° | Angle of the cord from vertical while the bob circles. |
| Disc mass (torsional) | kg | Mass of the disc. |
| Disc radius (torsional) | m | Radius of the disc. |
| Rod diameter (torsional) | mm | Diameter of the torsion rod. |
| Shear modulus of the rod (torsional) | GPa | Shear modulus of the rod material. |
Method and formulas
- Simple: T = 2π √(l ÷ g). Compound: T = 2π √(k₀² ÷ g r).
- Conical: T = 2π √(l cos θ ÷ g) for one revolution.
- Torsional: T = 2π √(J ÷ k) with J = M R² ÷ 2 and k = G π d⁴ ÷ 32 l (the handbook’s SI form T = 8 √(π M R² l ÷ d⁴ G)).
- Large swings: period × (1 + θ₀² ÷ 16), θ₀ in radians.
- Example: a 1 m simple pendulum has a period of 2.0064 s.
Limitations
- Small-angle formulas are about 1% low at 22° and 2% low at 32° amplitude; the correction shown covers moderate swings.
- Damping and pivot friction are neglected.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Measure a part’s moment of inertia by swinging it
- Check the natural frequency of a hanging load
- Design a torsion-rod timing or test device
Who it is for
- Design engineers
- Mechanical engineers
- Engineering students
What you get
- Period of one complete cycle
- Frequency
- Equivalent simple-pendulum length
Interpreting the result
- To find an unknown moment of inertia, time many swings and solve the compound formula for k₀.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 173–174. Topic reference; no source tables or recommended settings are reproduced.
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