The Final Angular Momentum Calculator helps students, physicists, and engineers find the final angular momentum of a rotating body. It’s a vital tool in analyzing spinning objects, rotating machinery, satellites, or even sports movements like figure skating spins. This calculator works for simple fixed-shape systems or for cases where the object changes shape or mass distribution. Understanding final angular momentum helps design safe mechanical systems, check stability, and apply conservation principles accurately. It belongs to the Rotational Motion and Dynamics Calculator category and supports education, research, and engineering design.
formula of Final Angular Momentum Calculator
1. For a single rotating object:
Final Angular Momentum (L) = I × ω
Where:
L = final angular momentum (kg·m²/s)
I = moment of inertia (kg·m²) — depends on the object’s mass and shape
ω = final angular velocity (radians per second)
2. If the moment of inertia changes:
Use Conservation of Angular Momentum:
L_initial = L_final
So,
I_initial × ω_initial = I_final × ω_final
Then, solve for final angular velocity:
ω_final = (I_initial × ω_initial) / I_final
Finally,
L_final = I_final × ω_final
This applies when objects change configuration, like a diver tucking in during a spin.
Common Final Angular Momentum Reference Table
This table shows typical moments of inertia for common shapes and simple final angular momentum examples.
| Object Shape | Moment of Inertia (I) Formula | Example L (at ω = 5 rad/s) |
|---|---|---|
| Solid sphere (radius R) | I = (2/5) m R² | 2 kg sphere, 0.5 m → L ≈ 1 kg·m²/s |
| Solid cylinder (axis) | I = (1/2) m R² | 3 kg cylinder, 0.3 m → L ≈ 0.675 kg·m²/s |
| Thin rod (center) | I = (1/12) m L² | 1 kg rod, 1 m → L ≈ 0.417 kg·m²/s |
Check these to get an idea for simple cases.
Example of Final Angular Momentum Calculator
Let’s solve a step-by-step example.
Example 1: Fixed Shape
A solid disk of mass 4 kg and radius 0.2 m spins at a final angular speed of 10 rad/s.
Moment of Inertia for disk (rotating about center):
I = (1/2) m R²
= (1/2) × 4 × (0.2)²
= (1/2) × 4 × 0.04
= 0.08 kg·m²
Final Angular Momentum:
L = I × ω
= 0.08 × 10
= 0.8 kg·m²/s
Example 2: Changing Shape
A skater spins with I_initial = 5 kg·m² at 2 rad/s, then pulls arms in to I_final = 2 kg·m².
ω_final = (I_initial × ω_initial) / I_final
= (5 × 2) / 2
= 5 rad/s
L_final = I_final × ω_final
= 2 × 5
= 10 kg·m²/s
Most Common FAQs
It helps describe and predict the motion of rotating systems, from wheels and engines to planets and satellites.
In a closed system with no external torque, yes. This is the Conservation of Angular Momentum.
Yes, but add up individual angular momenta for each part. For complex systems, use advanced dynamics methods.