BIZOMAT MAKEOS · Mechanics
Mass Moment of Inertia
What is the mass moment of inertia of this body about the chosen axis?
Mass moment of inertia and radius of gyration of common solids, with the parallel-axis transfer to an offset axis, for drive sizing and acceleration.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Body and axis | choice | The body and the axis it rotates about. |
| Mass | kg | Mass of the body. |
| Dimension A (outer diameter, or side a, or rod length) | mm | Outer diameter (disc, ring, cylinder, sphere), side a (block) or length (rod). |
| Dimension B (inner diameter, length, or side b) | mm | Inner diameter (ring), length (cylinder across), side b (block); not used for disc, sphere or rod. |
| Offset of rotation axis from centre (parallel axis) | mm | Distance between the body’s own central axis and the actual (parallel) rotation axis; 0 if it rotates about its centre. |
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: Body and axis
- What it is
- The idealised body shape and the axis it turns about, which decides the formula and what dimensions A and B mean.
- Units and format
- Choose one of 6 options from the list: Solid cylinder or disc, about its own axis; Hollow cylinder, about its own axis; Solid cylinder, about a central axis across it; Rectangular block, about a central axis normal to face a × b; Solid sphere, about a diameter; Slender rod, about one end.
- Example
- The form opens with “Solid cylinder or disc, about its own axis” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a complex part, CAD mass properties are more accurate than any of these idealised shapes.
What this means: Mass
- What it is
- The mass of the body, by weighing or from the CAD model. The tool assumes it is spread evenly through the idealised shape.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 10000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The moment of inertia rises in proportion to the mass.
- Get it right
- Holes and pockets change both the mass and how it is distributed.
What this means: Dimension A (outer diameter, or side a, or rod…
- What it is
- The main size of the body: the outer diameter of a disc, ring, cylinder or sphere, side a of a block, or the full length of a rod.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The moment of inertia rises with the square of this dimension.
- Get it right
- Enter the diameter, not the radius, for round shapes; the tool halves it.
What this means: Dimension B (inner diameter, length, or side b)
- What it is
- The second size of the body: the inner diameter of a ring, the length of a cylinder turning about an axis across it, or side b of a block.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Not used for a disc, sphere or rod. For a ring it must be smaller than the outer diameter.
What this means: Offset of rotation axis from centre (parallel a…
- What it is
- The distance between the body’s own axis and the actual rotation axis when the two are parallel, for a body that turns off-centre. Zero if it turns about its own axis.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds the mass times the square of the offset to the moment of inertia.
- Get it right
- The parallel-axis rule is exact only from an axis through the centre of mass; for the rod, whose formula is already about its end, leave it at 0.
Method and formulas
- Disc ½mr²; ring ½m(rₒ² + rᵢ²); cylinder across m(3r² + L²) ÷ 12; block m(a² + b²) ÷ 12; sphere ⅖mr²; rod about end mL² ÷ 3.
- Parallel axis: J = J_c + m d².
- Radius of gyration k = √(J ÷ m).
- Example: a 10 kg, 300 mm disc has J = 0.1125 kg·m².
Limitations
- Uniform density and the idealised shape are assumed; holes and features change the result.
- The rod formula already refers to the end axis; the offset is added from that axis.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a motor for a rotary table, roller or disc
- Find inertia of a part about a pivot
- Compare inertia of design options
Who it is for
- Automation engineers
- Design engineers
What you get
- Moment of inertia about the body axis
- Moment of inertia about the offset axis
- Radius of gyration
Interpreting the result
- Use the inertia with the motor’s rotor inertia and any gear ratio to check acceleration.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 228–240. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Moments of inertia of machine parts
In Inertia and Flywheels by the Numbers
What it covers: How to calculate the moment of inertia and radius of gyration of discs, rings, tubes, blocks and offset parts, and why the mass far from the axis is the mass that counts.
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