Angular Momentum Conservation

How to Use This Angular Momentum Conservation Calculator

This tool lets you explore conservation of angular momentum for a rigid body rotating about a fixed axis. It uses the relation

I₁ ω₁ = I₂ ω₂

• Enter any three of the four quantities: initial moment of inertia I₁, initial angular velocity ω₁, final moment of inertia I₂, or final angular velocity ω₂.
• Leave the quantity you want to solve for blank.
• Click Compute to calculate the missing value while keeping angular momentum constant.
• Use consistent SI units: I in kg·m² and ω in rad/s.

What Is Angular Momentum Conservation?

When no net external torque acts on a system, its angular momentum stays constant in time. For a rigid body rotating about a fixed axis, the angular momentum L is the product of moment of inertia and angular velocity:

L = I ω

If the mass distribution changes (for example, a figure skater pulling in their arms), the moment of inertia changes and the angular velocity adjusts so that the product Iω remains the same. This gives the conservation equation used by the calculator:

I₁ ω₁ = I₂ ω₂.

This relation follows from the rotational form of Newton’s second law. Torque τ is the rotational analogue of force and equals the rate of change of angular momentum:

τ = .

When the net external torque is zero, dL/dt = 0 and L is constant in time.

Angular Momentum in MathML Form

The conservation relation can also be written in MathML, which some browsers and screen readers can interpret directly:

Here I₁ and I₂ are the initial and final moments of inertia, and ω₁ and ω₂ are the initial and final angular velocities.

The Equation I₁ω₁ = I₂ω₂ and Rearrangements

The calculator algebraically rearranges the conservation equation to solve for the unknown quantity. Starting from

I₁ ω₁ = I₂ ω₂,

you can solve for each variable:

• Solving for final angular velocity ω₂ (given I₁, ω₁, I₂):

  ω₂ = (I₁ / I₂) × ω₁

• Solving for initial angular velocity ω₁ (given I₁, I₂, ω₂):

  ω₁ = (I₂ / I₁) × ω₂

• Solving for final moment of inertia I₂ (given I₁, ω₁, ω₂):

  I₂ = (I₁ ω₁) / ω₂

• Solving for initial moment of inertia I₁ (given I₂, ω₁, ω₂):

  I₁ = (I₂ ω₂) / ω₁

The calculator automatically detects which field is blank and applies the appropriate rearranged formula.

Worked Example

Consider a simplified model of a figure skater spinning with arms extended. Suppose:

• Initial moment of inertia: I₁ = 4.0 kg·m²
• Initial angular velocity: ω₁ = 2.0 rad/s
• Final moment of inertia after pulling arms in: I₂ = 2.0 kg·m²
• Unknown: final angular velocity ω₂

Using conservation of angular momentum,

I₁ ω₁ = I₂ ω₂

Rearrange to solve for ω₂:

ω₂ = (I₁ / I₂) × ω₁

Plug in the numbers:

ω₂ = (4.0 / 2.0) × 2.0 rad/s = 2 × 2.0 rad/s = 4.0 rad/s

So by halving the moment of inertia, the skater doubles their angular speed, while angular momentum L stays constant at

L = I₁ ω₁ = 4.0 × 2.0 = 8.0 kg·m²/s.

Moment of Inertia Explained

The moment of inertia I measures how strongly an object resists changes in its rotational motion about a particular axis. It depends on how mass is distributed relative to that axis:

• For a point mass m at distance r from the axis: I = m r².
• For extended bodies, I is found by integrating r² over the mass distribution: I = ∫ r² dm.
• Bringing mass closer to the axis decreases I; spreading mass out increases I.

Because L = I ω must remain constant when no external torque acts, a decrease in I leads to an increase in ω, and vice versa. This trade-off underlies many rotational phenomena, from collapsing gas clouds that spin up as they contract to spinning satellites that use internal mass shifts for control.

Interpreting the Calculator Results

When you click Compute, the output gives the missing quantity that makes I₁ ω₁ equal to I₂ ω₂. Interpreting the result:

• If the final moment of inertia I₂ is smaller than I₁, the computed ω₂ will be larger than ω₁ (spin-up).
• If I₂ is larger than I₁, the computed ω₂ will be smaller than ω₁ (spin-down).
• If the calculator returns a very large or very small value, check that you did not mix units (for example, degrees/s instead of rad/s).
• A negative angular velocity corresponds to rotation in the opposite direction around the same axis.

Typical Use Cases and Comparison

The same conservation principle appears in many settings. The table below compares a few common scenarios where a simple I₁ ω₁ = I₂ ω₂ model is often reasonable.

Assumptions and Limitations

The calculator is intentionally simple. It is based on a one-line conservation law and is best used for introductory physics problems and idealized scenarios. Keep these assumptions and limitations in mind:

• No net external torque: The formula assumes the total external torque on the system is effectively zero during the motion being analyzed.
• Rigid body (or effectively rigid): The calculation treats the object as a rigid body rotating about a fixed axis, with well-defined initial and final moments of inertia.
• Fixed rotation axis: The direction of the rotation axis is assumed not to change. Precession and complex 3D rotations are not modeled.
• Instantaneous or step-like change in I: The tool assumes a clear “before” and “after” state with I₁ and I₂. It does not track continuous time evolution during the change.
• Consistent units: All inputs should use SI units: I in kg·m² and ω in rad/s. Mixing units (for example, g·cm² and rad/s, or degrees/s instead of rad/s) will give incorrect results.
• Scalar treatment of angular velocity: Only the magnitudes of angular velocities are considered; vector directions and 3D effects are not included.
• No relativistic or quantum effects: The model is classical. It does not handle quantum angular momentum quantization or relativistic rotations.
• Educational and exploratory use: The results are suitable for teaching, learning, and basic estimates, not for safety-critical engineering design.

If your situation involves large external torques, rapidly changing rotation axes, deformable bodies, or complex multi-body interactions, a more detailed rotational dynamics model is required and this simple conservation calculator will only provide a rough approximation.