Rigid Body Dynamics Calculator

Four modes covering the core topics of rotational mechanics. Select a shape, adjust dimensions, and explore moment of inertia, angular momentum, rolling without slipping, and rotational equilibrium with real-time SVG diagrams.

Visualization

Mode

Shape & Dimensions

Shape

Mass (kg)

Radius (m)

Parallel Axis Offset (d)0.00 m

Presets

Results

I (center of mass)

0.1000 kg·m²

Step-by-Step

1. Moment of Inertia (Solid Cylinder)

I = \frac{1}{2}mr^2 = 0.1000 \text{ kg}\cdot\text{m}^2

Reference Guide

Moment of Inertia

Moment of inertia measures resistance to angular acceleration. It depends on mass distribution relative to the rotation axis.

Shape	Formula
Solid Cylinder / Disk	$\frac{1}{2}mr^2$
Hollow Cylinder	$\frac{1}{2}m(r_1^2+r_2^2)$
Solid Sphere	$\frac{2}{5}mr^2$
Hollow Sphere	$\frac{2}{3}mr^2$
Rod (center)	$\frac{1}{12}mL^2$
Rod (end)	$\frac{1}{3}mL^2$
Ring / Point Mass	$mr^2$

I = I_{cm} + md^2 \quad \text{(Parallel Axis Theorem)}

Angular Momentum and Torque

Angular momentum is the rotational analogue of linear momentum. Torque causes angular acceleration.

L = I\omega

\alpha = \frac{\tau}{I}

KE_{rot} = \frac{1}{2}I\omega^2

Rolling Without Slipping

When an object rolls without slipping, the contact point is instantaneously at rest. The constraint v = rω links translational and rotational motion.

a = \frac{g\sin\theta}{1 + \frac{I}{mr^2}}

KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2

Objects with smaller I/(mr²) accelerate faster. A solid sphere beats a hollow sphere, which beats a ring, on the same incline.

Rotational Equilibrium

A rigid body is in rotational equilibrium when the net torque about any point is zero.

\sum \tau = 0

\tau = F \times d_{\perp}

Torques that tend to rotate counter-clockwise are positive. Clockwise torques are negative. When these sum to zero, the beam is balanced.

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