Tables almost always give mass moments of inertia for an axis through the centre of mass. If a body rotates about a different, parallel axis, the parallel-axis theorem gives the correct value: JA = JS + m·rS². The additional term m·rS² is always positive, which is why the inertia about the centroidal axis is the smallest of all parallel axes.
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Tables almost always give mass moments of inertia for an axis through the centre of mass. If a body rotates about a different, parallel axis, the parallel-axis theorem gives the correct value: JA = JS + m·rS². The additional term m·rS² is always positive, which is why the inertia about the centroidal axis is the smallest of all parallel axes.
A disc of mass m = 12 kg has JS = 0.08 kg·m² about its centroidal axis. Rotated instead about a parallel axis at a distance rS = 150 mm = 0.15 m, the parallel-axis theorem gives JA = 0.08 kg·m² + 12 kg · (0.15 m)² = 0.08 + 0.27 = 0.35 kg·m². The additional term is therefore more than three times the centroidal inertia itself. A second, easily verified example is the thin rod: about its midpoint JS = (1/12)·m·l², and about its end at a distance l/2 it follows that JA = (1/12)·m·l² + m·(l/2)² = (1/3)·m·l² – exactly the familiar table value for rotation about the rod end.
Valid for a rigid body and only between two parallel axes, one of which must pass through the centre of mass. Between two arbitrary parallel axes that both miss the centre of mass the formula does not apply – there you first reduce to the centroidal axis and continue from it. Non-parallel or inclined axes, products of inertia and principal axes in spatial rotation, elastic deformation of the body under centrifugal load, and assembled bodies whose individual parts each need their own distance are all excluded.
Formula collections almost always give mass moments of inertia for an axis through the centre of mass. In practice, however, a part often rotates about a different axis – a lever about its bearing, a flap about its hinge. The parallel-axis theorem converts the table value to that actual axis.
The mass moment of inertia J describes how strongly a body resists angular acceleration; for rotation it plays the role that mass plays for straight-line motion. Mechanical engineering usually writes it as J, physics often as I. The parallel-axis theorem – also called Steiner's theorem – links the value JS about an axis through the centre of mass S with the value JA about a parallel axis through an arbitrary point A: JA = JS + m·rS², where m is the total mass and rS the perpendicular distance between the two axes. The additional term m·rS² equals the inertia the entire mass would have if it were concentrated at the centre of mass orbiting A at distance rS.
A hammer held at its head turns easily. Held at the end of its handle, considerably more effort is needed, although it is the same hammer with the same mass. The Steiner term captures exactly that difference – and because the distance enters squared, it grows very quickly.
JA = JS + m · rS²
Parallel-axis theorem: JA = JS + m·rS²Reduction to the centre of mass: JS = JA − m·rS²Axis distance from two inertias: rS = √((JA − JS)/m)Thin rod about its midpoint: JS = (1/12)·m·l², about its end: JA = (1/3)·m·l²Solid cylinder about its own axis: JS = ½·m·r²Sphere about a centroidal axis: JS = (2/5)·m·r²Consequence: JA ≥ JS for every parallel axis| Symbol / input | Meaning |
|---|---|
| Moment of inertia JA about the target axis | The mass moment of inertia about the actual axis of rotation through point A. The mass moment of inertia describes how strongly a body resists angular acceleration – for rotation it plays the role that mass plays for straight-line motion. The letter J is customary in mechanical engineering; physics texts often use I for the same quantity. |
| Moment of inertia JS about the centroidal axis | The mass moment of inertia about the parallel axis through the body's centre of mass S. Source: a formula collection for the basic shape in question, for example JS = ½·m·r² for a solid cylinder about its axis or JS = (1/12)·m·l² for a thin rod about its midpoint. Important: the reference point S must be the centre of mass – the formula may not be used between two arbitrary parallel axes. |
| Mass m | The total mass of the body. Source: weighing, or computed from volume and density. Always enter the complete mass, even if only part of the body lies far from the axis of rotation. |
| Axis distance rS | The perpendicular distance between the centroidal axis and the actual axis of rotation. Both axes must be parallel – for non-parallel axes the parallel-axis theorem does not apply. Since rS enters squared, the additional term grows very quickly: a distance of just one cylinder radius already triples the inertia of a solid cylinder. |
JS is the mass moment of inertia about the parallel axis through the centre of mass, usually from a formula collection: for a solid cylinder about its axis JS = ½·m·r², for a thin rod about its midpoint JS = (1/12)·m·l², for a sphere JS = (2/5)·m·r². m is the total mass of the body. rS is the perpendicular distance between the centroidal axis and the actual axis of rotation.
First locate the body's centre of mass and confirm that the actual axis of rotation really is parallel to the centroidal axis. Then take JS from a formula collection for the matching basic shape, enter the total mass, and measure the axis distance or take it from the drawing. For assembled parts, treat each component separately with its own distance to the common axis and add the individual values afterwards. Conversely, if a measured JA is available – from a pendulum test, say – you can reduce it back to JS by selecting JS as the target.
A disc of m = 12 kg has JS = 0.08 kg·m² about its centroidal axis. About a parallel axis at rS = 150 mm this gives JA = 0.08 + 12·0.15² = 0.08 + 0.27 = 0.35 kg·m². The Steiner contribution is therefore more than three times the centroidal value. An example that can be checked fully against the table is the thin rod: about its midpoint JS = (1/12)·m·l². Moving the axis by l/2 to the rod end gives JA = (1/12)·m·l² + m·(l/2)² = (1/12 + 1/4)·m·l² = (1/3)·m·l² – exactly the table value for rotation about the rod end. For m = 6 kg and l = 2 m that means JS = 2 kg·m² and JA = 8 kg·m².
Because m·rS² is always positive, the inertia about the centroidal axis is always the smallest of all parallel axes – the theorem's most important qualitative statement. The quadratic influence of distance has concrete consequences: for a solid cylinder with JS = ½mr², an axis offset of just one radius already triples the inertia to 1.5mr². Drives for off-centre mounted masses therefore need considerably more acceleration torque than a rough calculation with the table value suggests. Conversely it is almost always more effective to move a rotating mass closer to the axis than to make it lighter.
The calculation runs in kilograms, metres and therefore kg·m² internally. The inertias may be entered in kg·m² or kg·cm², the mass in mg, g, kg, t, oz or lb, and the axis distance in mm, cm, m or inches. Note for hand calculations: the distance must be inserted in metres for the result to come out in kg·m² – a distance inserted in millimetres is wrong by a factor of one million.
Acceleration torque for off-centre mounted levers, flaps, swing arms and cams; pendulums and swing doors; inertia of assembled rotors made of discs, hubs and shafts; evaluating pendulum tests to determine an unknown inertia; converting manufacturer data to the actual installed position; estimating the rotating masses of robot axes and tool turrets.
Valid for a rigid body and only between two parallel axes, one of which must pass through the centre of mass. Between two arbitrary parallel axes that both miss the centre of mass the formula may not be applied directly: first reduce to the centroidal axis and then continue to the second axis. Inclined or intersecting axes, products of inertia and principal axes in spatial rotation, elastic deformation of the body under centrifugal load, and assembled bodies as a whole – whose parts must be handled individually and summed – are not covered.
Common mistake: The most serious error is applying the formula between two axes neither of which passes through the centre of mass; the result is then simply wrong. Almost as common is a unit error in the distance: millimetres instead of metres misses the result by a factor of 10⁶. A third error is neglecting the Steiner contribution at small distances – because of the square it already dominates once the distance approaches the size of the part. And finally the theorem must not be carried over to inclined axes; that requires products of inertia and a principal-axis transformation.
Acceleration torque for off-centre mounted levers, flaps, swing arms and cams; pendulums and swing doors; inertia of assembled rotors made of discs, hubs and shafts; evaluating pendulum tests to determine an unknown inertia; converting manufacturer data to the actual installed position; estimating the rotating masses of robot axes and tool turrets.
JS is the mass moment of inertia about the parallel axis through the centre of mass, usually from a formula collection: for a solid cylinder about its axis JS = ½·m·r², for a thin rod about its midpoint JS = (1/12)·m·l², for a sphere JS = (2/5)·m·r². m is the total mass of the body. rS is the perpendicular distance between the centroidal axis and the actual axis of rotation.
Valid for a rigid body and only between two parallel axes, one of which must pass through the centre of mass. Between two arbitrary parallel axes that both miss the centre of mass the formula may not be applied directly: first reduce to the centroidal axis and then continue to the second axis. Inclined or intersecting axes, products of inertia and principal axes in spatial rotation, elastic deformation of the body under centrifugal load, and assembled bodies as a whole – whose parts must be handled individually and summed – are not covered.