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Rotating Body Inertia & Kinetic Energy

Find rotational energy, inertia or angular-speed magnitude for a rigid body about a specified axis.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

A body with moment of inertia 2 kg·m² rotating at 3 rad/s has kinetic energy 9 J. Reversing the rotation to −3 rad/s leaves its energy at 9 J.

Rotational kinetic energy = ½Iω²

Worked example

Enter these known values and leave the other values blank.

Moment of inertia about the chosen axis
2 kg·m²
Angular velocity about that axis
3 rad/s
Rotational kinetic energy
9 J

Assumptions and limitations

  • Moment of inertia must be positive and must correspond to the chosen axis. This model describes a rigid body rotating about that axis.
  • Angular velocity may be signed. Energy depends on its square and cannot be negative. Zero angular velocity gives zero rotational energy.
  • Solving angular velocity from energy returns the nonnegative square-root branch. Energy alone cannot identify the direction of rotation.
  • With zero energy and zero angular velocity, inertia is not uniquely determined. Translational kinetic energy is not included.
  • The lb·ft² option uses pound mass, not pound force. The hertz option counts complete revolutions per second; one revolution is exactly 2π radians.

Common questions

Does doubling angular speed double the energy?

For fixed inertia, doubling angular speed multiplies rotational kinetic energy by four.

Can a negative rotation have positive energy?

Yes. The sign indicates rotation direction; squaring angular velocity gives the same positive energy for either direction.

References

Bookify requires positive moment of inertia and nonnegative rotational kinetic energy, while allowing signed angular velocity. Bookify uses the exact relationship one revolution = 2π radians instead of the captured rounded angular-speed factor. Pound-mass square feet use the exact international pound and square international foot, replacing a rounded inertia factor.

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