Parallel Axis Theorem Calculator
Applies the parallel-axis theorem to a rigid body. On this Parallel Axis Theorem page, changing an entry updates the result and visible checking path.
Describe the system state
Shifted-axis moment of inertia
Following I = I_cm + Md²
The worked case uses Center-of-mass inertia = 2 kg·m², Mass = 5 kg, Axis offset = 1 m. These values provide a reproducible example, and no unannounced unit conversion is applied to them.
Arrange I = I_cm + Md² symbolically before substitution. That order makes an inverted ratio, omitted exponent, or misplaced number easier to identify.
Read the conservation model first
Applies the parallel-axis theorem to a rigid body. In laboratory collision data, this relationship is meaningful only when the reference frame, direction convention, and units remain consistent.
The named fields are center-of-mass inertia, mass, axis offset. Each belongs in a defined position within I = I_cm + Md²; writing values beside the symbols helps catch a transposition.
For parallel axis theorem, shifted-axis moment of inertia is treated as a nonnegative magnitude. If an entered combination produces a negative value, revisit the physical domain instead of reading the sign as a direction.
Reading shifted-axis moment of inertia in context
The calculator reports shifted-axis moment of inertia in kg·m². If that number enters a later formula, retain guard digits until the final operation.
Compare shifted-axis moment of inertia with the scale of the parallel axis theorem scenario. A metric-prefix mistake or inconsistent time unit can produce tidy arithmetic that is physically implausible.
For reproducibility, record center-of-mass inertia, mass, axis offset, their units, the reference direction, and I = I_cm + Md² rather than recording only the final numeral.
Test the result against the system boundary
Start the dimensional check with I = I_cm + Md². After cancellation, the surviving dimension should align with kg·m²; a mismatch means the setup needs correction.
Then change one input by a controlled amount and predict how shifted-axis moment of inertia ought to respond before recalculating. Direction and sensitivity provide separate checks on the arithmetic.
Boundaries of the simplified result
The Parallel Axis Theorem relationship uses the stated rotation axis and mass distribution. Deformation, bearing loss, shifting mass, or an unlisted external torque can change shifted-axis moment of inertia.
The precision of shifted-axis moment of inertia is limited by the least reliable measurement. Extra displayed digits enable verification, but safety-critical work needs validated data and a suitable engineering procedure.
A sensible next calculation
Useful follow-up calculations include flywheel stored energy calculator and radius of gyration calculator.
Continue with the page that matches the next system state, not merely one that repeats an input. Here, that choice follows from the parallel axis theorem result.
Common questions about the calculation
What does the shifted-axis moment of inertia represent?
It is shifted-axis moment of inertia under I = I_cm + Md² and the field definitions printed on this page.
How can the Parallel Axis Theorem result be checked?
Rearrange I = I_cm + Md² to recover one entered quantity, then confirm that the remaining unit is kg·m².
Do these inputs need consistent units?
Yes. Match every value to the unit beside its field before using I = I_cm + Md².
Why could another shifted-axis moment of inertia differ?
Gravity choice, rounding, sign conventions, reference frames, or different assumptions can shift the reported shifted-axis moment of inertia.