Library / Mechanical / MEC-19
Mechanical
Angular acceleration from a moment (Euler’s equation)
How fast a rigid body’s spin rate changes under a net moment about its centre of mass.
- Ref
- MEC-19
- Rev
- v1
- Field
- Mechanical
- Result
- α [rad/s²]
Planar motion of a rigid body of constant mass. Moments are taken about the centre of mass G, about the axis perpendicular to the plane. The companion force equation ΣF = maG gives the motion of G.
The net twisting effect divided by the body’s rotational inertia gives how quickly it spins up.
Calculator
- Time to reach 100 rad/s from rest5 s
- Angular acceleration191 rpm/s
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.
- Take moments about G, or about a fixed point, but do not mix the two without correction terms.
- The force equation and the moment equation are independent and are solved together.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| IG Moment of inertia | 0.001 to 100 kg·m² | 0.18 | 18,000 | 18,000 | +10% | −9.09 % |
| ΣMG Net moment about G | 0.1 to 1,000 N·m | 0.1111 | 1,111 | 1,111 | +10% | +10 % |
Sweep of Net moment about G
| Net moment about G (N·m) | Angular acceleration (rad/s²) |
|---|---|
| 0.1 | 0.1111 |
| 125.1 | 139 |
| 250.1 | 277.9 |
| 375.1 | 416.7 |
| 500.1 | 555.6 |
| 625 | 694.5 |
| 750 | 833.4 |
| 875 | 972.2 |
| 1,000 | 1,111 |
Sweep one variable
Variables and how each one contributes
Angular acceleration
What the equation gives you. Everything else feeds it.
Net moment about G
Net moment about the centre of mass. Directly proportional.
Moment of inertia
Inertia about the centre of mass. Inversely proportional.
Worked example with the values above
- ΣMG = 18 N·m; IG = 0.9 kg·m².
- Divide: 18 / 0.9.
- α = 20 rad/s².
Assumptions and limits
- Rigid body, constant mass.
- Planar motion.
Common mistakes
- Taking moments about a point that is accelerating.
- Omitting a couple.
Statistics and uncertainty
Both the moment and the inertia are usually estimated, and their relative errors add.
Where it is used
- Turret and gimbal drive sizing.
- Vehicle pitch and yaw dynamics.
Maths you need
- Newton’s second law: F = ma
- Moment: force × distance