Rotational Motion Simulator

1. Real‑World Phenomenon

Rotating machinery pervades modern technology. From electric motors spinning washers to reaction wheels orienting satellites, the ability of a torque to change angular speed sits at the heart of countless devices. Many introductory calculators merely rearrange algebraic kinematics equations, returning final angles or velocities without revealing how motion develops in time. This simulator keeps those analytic relations but augments them with a live model of a wheel responding to a constant torque. As the user adjusts initial angular velocity $\mathrm{\omega ₀}$, angular acceleration $\alpha$, or moment of inertia $I$, the wheel on the canvas speeds up or slows down accordingly. Below the canvas, striped bars show rotational kinetic energy and the work done by the applied torque, allowing learners to confirm that energy grows quadratically with speed. Being able to pause, reset, and export data turns the once static topic into an interactive experiment where the interplay of torque, inertia, and angular displacement is tangible.

2. Variables and Assumptions

The model assumes a rigid wheel of moment of inertia $I$ rotating about a fixed axis. The state at any instant is defined by angular position $\theta$ and angular velocity $\omega$. The user specifies initial conditions $\mathrm{\theta ₀},\mathrm{\omega ₀}$ along with constant angular acceleration $\alpha$, effectively representing the ratio of applied torque $\tau$ to inertia ($\alpha =\frac{\tau }{I}$). Gravitation and friction are ignored so the wheel spins freely. The numerical scheme integrates in SI units: radians, seconds, kilograms, and meters. Any non‑finite input is rejected, inertia must remain positive, and the time step is clamped between 0.001 and 0.1 s to maintain stability. Because the acceleration is constant, the system admits a closed‑form solution, but the simulation intentionally uses discrete time stepping to highlight numerical techniques useful when analytic expressions are unavailable.

3. Governing Equations

With constant angular acceleration the kinematic relations mirror those of linear motion. Angular position evolves as $\mathrm{\theta (t)}=\mathrm{\theta ₀}+\mathrm{\omega ₀t}+\frac{1}{2}\mathrm{\alpha t²}$, angular velocity as $\mathrm{\omega (t)}=\mathrm{\omega ₀}+\mathrm{\alpha t}$. Rotational kinetic energy is $K=\frac{1}{2}I\mathrm{\omega ²}$, while work performed by a constant torque equals $W=\tau \mathrm{\Delta \theta }=\mathrm{I\alpha }\mathrm{\Delta \theta }$. For perfect numerics, $W$ matches $K$ when starting from rest. Tracking both reveals energy drift introduced by finite time steps or round‑off. The simulator accumulates work by integrating $\tau \omega$ over time, a procedure that generalizes to variable torque.

4. Numerical Scheme

The animation advances using a second‑order Runge–Kutta (midpoint) method, striking a balance between simplicity and stability. Given current state $(\theta ,\omega )$ at time $t$, the algorithm computes preliminary slopes $\mathrm{k₁^\theta }=\omega$, $\mathrm{k₁^\omega }=\alpha$. A midpoint estimate updates to $\mathrm{\theta *}=\theta +\frac{1}{2}\mathrm{k₁^\theta \Delta t}$ and $\mathrm{\omega *}=\omega +\frac{1}{2}\mathrm{k₁^\omega \Delta t}$. The slopes at the midpoint $\mathrm{k₂^\theta }=\mathrm{\omega *}$, $\mathrm{k₂^\omega }=\alpha$ then update the state: $\theta \leftarrow \theta +\mathrm{k₂^\theta \Delta t}$ and $\omega \leftarrow \omega +\mathrm{k₂^\omega \Delta t}$. This method exactly reproduces constant‑acceleration motion while remaining robust for slowly varying torques if the simulator is extended. Energy and work increment according to the updated angular velocity and displacement. Because the midpoint method is symplectic for constant acceleration, energy error remains small even for relatively coarse steps, but the interface warns users when $\mathrm{\Delta t}$ grows so large that drift becomes visible in the energy bars.

5. Worked Example

Imagine a wheel with moment of inertia 2 kg·m² initially at rest. A motor applies a constant torque of 4 N·m, giving angular acceleration $\alpha =2$ rad/s². Integrating for 5 s with $\mathrm{\Delta t}=0.01$ s yields a final angular velocity of 10 rad/s and an angular displacement of 25 rad. Rotational kinetic energy climbs to $\frac{1}{2}I\mathrm{\omega ²}=100$ J, exactly matching the work $\mathrm{\tau \Delta \theta }=4\times 25$. Running the simulator with these inputs reproduces the acceleration curve; the energy bars rise together, and the text summary reports negligible difference between work and energy. If the time step is increased to 0.05 s, the final velocity remains close, but the kinetic energy lags slightly behind the work due to integration error, a discrepancy quantified by the numeric readout.

6. Comparison Table

The table contrasts the baseline case with two variants: a heavier wheel and a stronger torque. Values are sampled after 5 s.

A heavier wheel spins more slowly for the same torque, storing less energy, while doubling the torque doubles both the final speed and the accumulated work, quadrupling kinetic energy. The simulator visualizes these trends instantaneously.

7. How to Read the Animation

The canvas centers a wheel with a single spoke. As time advances, the spoke rotates according to the current angle. The animation rescales automatically to stay crisp on any device. Blue striped bars quantify rotational kinetic energy; orange bars show the mechanical work delivered by the constant torque. Their difference reflects numerical error, narrated in the aria‑live region for screen‑reader users. Keyboard users can focus the canvas and press the space bar to toggle play and pause. Tooltips on the inputs display units, and the text fallback summarizes the wheel's angle, speed, and energies so information does not rely solely on color or visuals.

8. Limitations

The model ignores friction, air drag, and structural flexing. Real motors cannot supply perfectly constant torque, and bearings introduce resistive torques that eventually halt the motion. The wheel is treated as a rigid body; deformable rotors would store additional elastic energy. Numerical integration with large time steps can accumulate error, leading to mismatches between work and kinetic energy. Users should interpret results within these idealizations.

9. Suggested Extensions

Future versions could include velocity‑dependent damping, step‑wise torques mimicking electric motor control, or even gravitational potential for rotating platforms. A phase‑space plot of $\omega$ versus $\theta$ would highlight the linear trajectories characteristic of constant acceleration. Allowing piecewise torque inputs would transform the tool into a simple rigid‑body dynamics playground useful for robotics and astronomy courses.

10. References and Related Tools

Key texts include H. Goldstein’s Classical Mechanics and J. L. Meriam & L. G. Kraige’s Engineering Mechanics: Dynamics. For related explorations, see the Rotational Kinetic Energy Calculator, experiment with collisions using the Elastic Collision Simulator, or compare translational motion in the Newton’s Second Law Simulator.