Library / Mechanical / MEC-16

Mechanical

Parallel-axis theorem

Moment of inertia about an axis parallel to one through the centre of mass.

Ref: MEC-16
Rev: v1
Field: Mechanical
Result: I_O [kg·m²]

I_{O} = I_{G} + m d^{2}

Axes must be parallel with one through the centre of mass G; d is the perpendicular offset between them. Units kg m². The definition is I_G = ∫r⊥² dm.

Spinning a body about an axis away from its centre of mass makes it harder to turn than spinning it about its own centre.

Calculator

Solve for

I_OMoment of inertia about OComputed

kg·m²

I_GInertia about G

kg·m²

0.001100

mMass

0.11,000

dAxis offset

0.013

Offset term m d²0.45 kg·m²
Radius of gyration about O0.4123 m

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Moment of inertia about O (kg·m²) current value 0.85 kg·m²

I_GInertia about GKey driver

0.451100

mMass

0.40990.4

dAxis offset

0.40145.4

I_G is the smallest possible moment of inertia of a body for axes in that direction.
Moving an axis out by d adds m d² regardless of the shape.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kg·m²)	Local step	Change in result
I_G Inertia about G	0.001 to 100 kg·m²	0.451	100.5	100	+10%	+4.71 %
m Mass	0.1 to 1,000 kg	0.409	90.4	89.99	+10%	+5.29 %
d Axis offset	0.01 to 3 m	0.4005	45.4	45	+10%	+11.1 %

Sweep of Axis offset

Axis offset (m)	Moment of inertia about O (kg·m²)
0.01	0.4005
0.3838	1.136
0.7575	3.269
1.131	6.799
1.505	11.73
1.879	18.05
2.252	25.77
2.626	34.89
3	45.4

Sweep one variable

Vary

Moment of inertia about O (kg·m²)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

I_O

Moment of inertia about O

Result · kg·m²

What the equation gives you. Everything else feeds it.

I_G

Inertia about G

kg·m² · typical 0.001 to 100

Inertia about the centre-of-mass axis. Adds directly.

Mass

kg · typical 0.1 to 1,000

Mass of the body. Scales the offset term.

Axis offset

m · typical 0.01 to 3

Perpendicular distance between the axes. Squared: the strongest driver at large offsets.

Worked example with the values above

Offset term: m d² = 5 × 0.3² = 0.45 kg·m².
Add I_G = 0.4.
I_O = 0.85 kg·m².

Assumptions and limits

Parallel axes, one through G.
Rigid body of constant mass.

Common mistakes

Using d between two axes that do not pass through G.
Forgetting to square d.

Statistics and uncertainty

Offset is squared, so a 1 cm error on 30 cm is about 7% on the offset term.

Where it is used

Pendulum and compound-body dynamics.
Gun and radar mounts.

Maths you need

Moment of inertia: kg m²
Squares: d²

Related formulas

Practise this formula