Library / Mechanical / MEC-18
Mechanical
Angular impulse and momentum (fixed axis)
Final spin rate of a body acted on by a steady moment about a fixed axis.
- Ref
- MEC-18
- Rev
- v1
- Field
- Mechanical
- Result
- ω₂ [rad/s]
For planar fixed-axis rotation HO,z = IOω and the angular impulse equals the change in H. This form assumes a constant moment M over the interval Δt.
A steady turning force applied for some time changes the spin rate by an amount set by how hard it is to turn the body.
Calculator
- Change in spin rate12 rad/s
- Final speed152.8 rpm
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.
- Doubling the inertia halves the spin-up for the same moment and time.
- The angular analogue of F Δt = m Δv.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| I Moment of inertia | 0.01 to 100 kg·m² | 4.3 | 3,004 | 3,000 | +10% | −6.82 % |
| M Moment | 0.1 to 500 N·m | 4.08 | 404 | 399.9 | +10% | +7.5 % |
| Δt Duration | 0.1 to 60 s | 4.6 | 364 | 359.4 | +10% | +7.5 % |
| ω₁ Initial spin rate | 0 to 100 rad/s | 12 | 112 | 100 | +10% | +2.5 % |
Sweep of Duration
| Duration (s) | Final spin rate (rad/s) |
|---|---|
| 0.1 | 4.6 |
| 7.587 | 49.52 |
| 15.07 | 94.45 |
| 22.56 | 139.4 |
| 30.05 | 184.3 |
| 37.54 | 229.2 |
| 45.02 | 274.1 |
| 52.51 | 319.1 |
| 60 | 364 |
Sweep one variable
Variables and how each one contributes
Final spin rate
What the equation gives you. Everything else feeds it.
Moment
Constant moment about the axis. Raises ω₂ in proportion.
Duration
Time the moment acts. Raises ω₂ in proportion.
Moment of inertia
Inertia about the axis. A heavier rotor speeds up less.
Initial spin rate
Spin rate before the moment acts. Adds directly.
Worked example with the values above
- Angular impulse: M Δt = 15 × 2 = 30 N·m·s.
- Spin-rate change: 30 / 2.5 = 12 rad/s.
- ω₂ = 4 + 12 = 16 rad/s.
Assumptions and limits
- Constant moment.
- Fixed axis through O.
Common mistakes
- Using rpm for ω.
- Forgetting the initial spin rate.
Statistics and uncertainty
If M varies during the interval, use its time average; timing errors scale ω₂ directly.
Where it is used
- Spin-up of rotors and flywheels.
- Gun traverse and turret drives.
Maths you need
- Moment of inertia: kg m²
- Angular momentum: H = Iω