Moment of Inertia of a System of Particles: I = Sum of m r Squared
The moment of inertia of a few point masses about an axis.
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How this calculator works
Choose whether you give each particle by its distance from the axis or by its x and y coordinates.
Enter the mass and the position of each particle. Leave the mass at 0 for a particle you do not use.
The calculator gives each m r squared, the total, the radius of gyration, the centre of mass, and the rotational energy and angular momentum if you give a spin rate.
Formulas used
- I = \sum m_i r_i^2 = m_1r_1^2 + m_2r_2^2 + \cdots + m_nr_n^2
- r_i^2 = x_i^2 + y_i^2
- I = MK^2 \Rightarrow K = \sqrt{I/M}
- E_k = \tfrac12 I\omega^2
Worked example
With the default numbers in the calculator above:
- Particle 1: mass times distance squared: 0.50 kg m2
- Particle 2: mass times distance squared: 3.00 kg m2
- Particle 3: mass times distance squared: 4.00 kg m2
- Total mass: 6.00 kg
- Moment of inertia of the system about the axis: 7.50 kg m2
- Radius of gyration: 1.12 m
- Angular speed: 6.28 rad/s
- Rotational kinetic energy: 148.04 J
- Angular momentum: 47.12 kg m2/s
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: How do you give the position of each particle? By its x and y coordinates (the axis is the z axis through the origin); Particle 1: mass 2 kg; Particle 1: x coordinate 3 m; Particle 1: y coordinate 4 m; Particle 2: mass 1 kg; Particle 2: x coordinate 1 m; Particle 2: y coordinate 0 m; Particle 3: mass (0 = not used) 0 kg; Particle 4: mass (0 = not used) 0 kg.
- Particle 1: mass times distance squared: 50.00 kg m2
- Particle 2: mass times distance squared: 1.00 kg m2
- Total mass: 3.00 kg
- Moment of inertia of the system about the axis: 51.00 kg m2
- Radius of gyration: 4.12 m
- Centre of mass: (2.33, 2.67) m
- Moment of inertia about a parallel axis through the centre of mass: 13.33 kg m2
- Angular speed: 6.28 rad/s
- Rotational kinetic energy: 1006.70 J
- Angular momentum: 320.44 kg m2/s
Common questions
What is the moment of inertia of a system of particles?
The sum of each mass times the square of its perpendicular distance from the axis. It is the rotational equivalent of mass.
What is the radius of gyration?
The distance from the axis at which the whole mass could be put to give the same moment of inertia, so I equals the mass times K squared.
What is the unit of moment of inertia?
Kilogram metre squared (kg m2) in SI, and gram centimetre squared (g cm2) in cgs. Its dimensional formula is M L2.
Does the answer depend on the axis?
Yes. The same body has a different moment of inertia about a different axis, because the distances change.
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Further reading: Any physics or engineering mechanics text (moment of inertia of a rigid body and radius of gyration).
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