This interactive simulation helps you visualize how mass distribution affects rotation and linear acceleration. It combines a Moment of Inertia Explorer (3D object inspector), a Ramp Race (rolling without slipping), Parallel Axis Theorem (I vs d graph), and a Torque & Angular Acceleration experiment with live F, τ, ω overlay.

Use the Object Inspector to choose shapes (solid sphere, hollow sphere, cylinder, hoop, box, cone, octahedron), adjust mass and radius, and see I and its formula. Race two objects down a ramp and compare K_trans vs K_rot and v_x, v_y vs time. Explore I = I_cm + Md² with the I vs d graph, and apply a force to spin objects via τ = r F sin θ; use θ ∈ [−180°, 180°] to accelerate or decelerate rotation.

 

Math behind the Simulation

1. Moment of Inertia (I)

The moment of inertia quantifies resistance to angular acceleration about a given axis. For shapes with symmetry, I can be written as I = f M R², where f depends only on geometry and R is a characteristic size. About a symmetry axis through the center of mass:

• Solid sphere: I = (2/5) M R²
• Hollow sphere: I = (2/3) M R²
• Solid cylinder (disk): I = (1/2) M R²
• Hoop (ring): I = M R²
• Box (cube), R = half-side: I ≈ (2/3) M R²
• Cone (R = base radius): I = (3/10) M R²
• Octahedron (R = vertex distance): I ≈ (1/5) M R²

For box, cone, and octahedron, R is defined as above so that the same rolling formula a = g sin θ / (1 + f) applies when comparing shapes.

2. Rolling without slipping on a ramp

Linear acceleration down the ramp: a = g sin θ / (1 + I/(MR²)). The factor f = I/(MR²) depends only on shape (e.g. 1/2 for a cylinder, 1 for a hoop). Lower f ⇒ larger a ⇒ that object wins the race. Energy splits into K_trans = ½ M v² and K_rot = ½ I ω², with ω = v/R when rolling without slipping. The race runs until both objects reach the bottom; the ramp overlay plots v_x and v_y vs time for each.

3. Parallel Axis Theorem

If the rotation axis is parallel to the one through the center of mass (COM) and offset by distance d, then I = I_cm + M d². I grows with d², so moving the axis away increases the difficulty of rotation. The I vs d graph plots this parabola.

4. Torque and angular acceleration

Torque is τ = r × F. With r = lever arm, F = force magnitude, and θ = angle between r and F, the magnitude is τ = r F sin θ. The direction of τ follows the right-hand rule (thumb along τ, fingers from r toward F); the object rotates in that sense. Tangential force (θ = 90°) gives maximum torque; force along r (θ = 0°) gives zero. Angular acceleration α = τ / I.

The angle θ can vary in [−180°, 180°]. Opposite θ (e.g. +90° vs −90°) yields opposite torque and thus opposite rotation. If the object is already spinning, applying force in the same direction as ω accelerates it; applying in the opposite direction decelerates or reverses it. The lever arm r is constrained to the object radius R; when R is changed, r is set equal to R.

In Torque mode, an overlay plots force F, torque τ, and angular speed |ω| vs time. The ω curve is colored by rotation direction (e.g. ω+ vs ω−) so you can see when the object speeds up, slows down, or reverses.

 

Usage

Follow these steps to explore the Rotational Motion Lab:

1. View: Use the View dropdown to switch between Object Inspect and Race. Only one view is shown at a time.
2. Object Inspect: Choose Shape (solid sphere, hollow sphere, cylinder, hoop, box, cone, octahedron), M and R. The 3D view shows the object and the rotation axis; solid shapes use a striped texture so rotation is visible. Use Iso, Front, Top, Side and Zoom+/Zoom− to change the camera, or drag to orbit. Run/Stop toggles auto-rotation (including in Torque mode); Step Fwd/Step Bwd advance or reverse rotation when stopped. Reset keeps Shape and Mode but restores M, R, force settings, and rotation to defaults. Mode (Standard / Parallel Axis / Torque) adds d and I vs d graph, or F, r, θ and Apply Force. In Torque mode, r ≤ R and is set to R when R changes; θ ∈ [−180°, 180°] (default 90°). Hold Apply Force to spin; if Run is stopped, pressing it starts Run first, then applies force.
3. Race: Set Object A and Object B presets and Ramp °. Run/Stop toggles the race; objects roll without slipping. The run auto-stops when both reach the bottom. When stopped, Step Fwd and Step Bwd advance or reverse by a small timestep. Reset restores initial positions and energy bars. An overlay plots v_x, v_y vs time for each object; energy bars show K_trans vs K_rot.

Tips: Compare cylinder vs hoop, or box vs octahedron, in the race to see how shape affects acceleration. In Torque mode, use θ = 0° (force toward center) for zero torque and no spin; 90° (tangential) for maximum torque. Use opposite θ to decelerate or reverse an already spinning object. The torque overlay shows |ω| rising when you speed up; colors indicate rotation direction. In Parallel Axis mode, move d and watch the I vs d parabola.

 

Visualizations

• 3D Object Inspector: Renders the selected shape (solid with striped texture, or wireframe for hollow). A bright axis shows the rotation axis. In Parallel Axis mode, a dashed ghost axis marks the COM; a solid active axis shows the parallel axis. In Torque mode, a force arrow (green when effective, red when nearly radial) shows magnitude, lever arm, and angle. A top-right overlay plots F, τ, and |ω| vs time, with ω colored by rotation direction.
• Ramp Race: Side-view 2D ramp. Two objects (A cyan, B green) roll down; acceleration a is shown. A top-right overlay plots v_x, v_y vs time for each. A legend and live bars show K_trans (cyan) and K_rot (green).
• I vs d graph: Parabola I = I_cm + M d² with a moving dot at the current d (Parallel Axis mode).

Controls

• Shape, M, R: Object type (sphere, cylinder, hoop, box, cone, octahedron), mass (kg), radius (m). Updates I and the 3D view.
• Ramp °: Ramp angle (degrees) for the race.
• Mode: Standard / Parallel Axis / Torque.
• Run / Stop, Step Fwd / Step Bwd, Reset: Run/Stop toggles 3D auto-rotation (also in Torque mode). Step advances or reverses rotation when stopped. Reset (Inspect) keeps Shape and Mode; resets M, R, force, rotation.
• d (Parallel Axis): Offset of the rotation axis from the COM (m).
• F, r, θ (Torque): Force (N), lever arm (m) with r ≤ R and r = R when R changes, angle [−180°, 180°] (default 90°). Apply Force: hold to spin; starts Run if stopped.
• View: Object Inspect / Race. Only the selected view is shown.
• Object A / B, Run / Stop, Step Fwd / Step Bwd, Reset: Race presets and actions (Race view). Reset restores positions and energy bars.