The formula
- System of point masses
I = Σ mᵢrᵢ²- Solid disk or cylinder about its central axis
I = ½mR²- Thin rod about an axis through its center
I = (1/12)mL²- Parallel-axis theorem
I = I_cm + md²
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
I | Moment of inertia about the stated axis | kg·m² |
mᵢ | Mass of one point particle in the system | kg |
rᵢ | Distance of that particle from the axis | m |
m | Total mass of the body | kg |
R | Radius of the disk or cylinder | m |
L | Length of the rod | m |
I_cm | Moment of inertia about a parallel axis through the center of mass | kg·m² |
d | Distance between the two parallel axes | m |
When it applies
- You need the rotational counterpart of mass, either for Στ = Iα or for rotational kinetic energy K = ½Iω².
- Name the axis before quoting a value. A rod spun about its center has one quarter the moment of inertia of the same rod spun about one end.
- Use the parallel-axis theorem only when the new axis is parallel to one through the center of mass, a distance d away.
Worked example
Problem. A solid disk of mass 2.5 kg and radius 0.20 m spins about its central axis. Find its moment of inertia, then the angular acceleration a 0.75 N·m torque would produce.
- For a solid disk about its central axis, I = ½mR² = ½(2.5 kg)(0.20 m)².
- (0.20 m)² = 0.040 m², so I = (0.5)(2.5)(0.040) = 0.050 kg·m².
- Apply the rotational form of Newton's second law: α = τ / I = 0.75 N·m / 0.050 kg·m².
- α = 15 rad/s².
Answer. I = 0.050 kg·m², and the disk gains angular speed at 15 rad/s².
Common mistakes
- Quoting a moment of inertia without naming the axis, which leaves the number meaningless.
- Forgetting to square the radius or the length, which is where most of the arithmetic slips happen.
- Using the parallel-axis theorem from an axis that is not through the center of mass. It only works starting from the center-of-mass axis.
- Adding moments of inertia taken about different axes. They add only when they share the same axis.
Related formulas
- Torque formula:
τ = rF·sin θ - Momentum formula:
p = mv - Centripetal force formula:
F_c = mv² / r