BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Pendulum
- What it is
- The kind of pendulum: a point mass on a cord, a rigid body swinging about a pivot, a bob moving in a horizontal circle, or a disc twisting on a slender rod. Each type uses its own inputs.
- Units and format
- Choose one of 4 options from the list: Simple (point mass on a cord); Compound (rigid body); Conical (bob circling); Torsional (disc on a rod).
- Example
- The form opens with “Simple (point mass on a cord)” selected as a worked example. Choose the option that matches your case.
What this means: Length (cord, or rod for torsional)
- What it is
- For simple and conical pendulums, the cord length from the pivot to the centre of the bob; for a torsional pendulum, the free length of rod that twists between its clamp and the disc. Not used for a compound pendulum.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 10000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The period of a simple pendulum rises with the square root of length: four times the length doubles it.
- Get it right
- Measure to the centre of the bob, not to its lowest point.
What this means: Swing amplitude (simple, compound)
- What it is
- How far the pendulum swings either side of vertical, used to correct the period of simple and compound pendulums for larger swings. It has no effect on the conical or torsional types.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The corrected period is the small-swing period × (1 + θ₀² ÷ 16), with θ₀ in radians.
- Get it right
- Use the angle from vertical to one extreme, not the whole swing from side to side.
What this means: Radius of gyration about the pivot (compound)
- What it is
- For a compound pendulum, the distance from the pivot at which the body’s whole mass could be concentrated to give the same moment of inertia about the pivot: √(moment of inertia about the pivot ÷ mass).
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 1000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.6 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The equivalent length is k₀² ÷ r, so the period rises in proportion to k₀.
- Get it right
- It is taken about the pivot, not about the centre of gravity, so it cannot be smaller than the pivot-to-centre-of-gravity distance.
What this means: Pivot to centre of gravity (compound)
- What it is
- For a compound pendulum, the straight-line distance from the pivot axis to the body’s centre of gravity.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 1000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.4 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger distance shortens the equivalent length k₀² ÷ r and so the period.
- Get it right
- Find the centre of gravity by balancing the part on a knife edge or from the CAD mass properties.
What this means: Cord angle from vertical (conical)
- What it is
- For a conical pendulum, the steady angle the cord makes with the vertical while the bob circles. The period given is the time for one full revolution.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 89°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger angle shortens the period, which depends on l cos θ.
What this means: Disc mass (torsional)
- What it is
- For a torsional pendulum, the mass of the disc hanging on the rod. The disc is treated as a uniform solid disc.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 1000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The period rises with the square root of the disc mass.
What this means: Disc radius (torsional)
- What it is
- For a torsional pendulum, the outer radius of the disc, from its centre on the rod axis to its rim.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 100 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The period rises in proportion to the radius, because the disc inertia goes with R².
- Get it right
- Radius, not diameter: halve a measured diameter.
What this means: Rod diameter (torsional)
- What it is
- For a torsional pendulum, the diameter of the slender rod that twists, taken over its free length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The period falls with the square of the rod diameter.
- Get it right
- Measure it carefully: the rod stiffness depends on the fourth power of this diameter.
What this means: Shear modulus of the rod (torsional)
- What it is
- For a torsional pendulum, the shear modulus (modulus of rigidity) of the rod material, which sets how much torque it takes to twist the rod.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 79.3 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The period falls with the square root of the shear modulus.
- Get it right
- Use the shear modulus G, not the tensile modulus E, from the material data sheet, and enter it in GPa.
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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Open the Pendulum Period: Simple, Compound, Conical and Torsional calculator