Planar Rigid-Body Motion

Planar motion occurs when all particles of a rigid body move along paths that maintain a constant distance from a specified plane. This fundamental concept allows us to understand how objects move in space while staying within a two-dimensional reference frame. Every point on the body follows a trajectory confined to this plane, making planar motion distinct from three-dimensional motion.

Rectilinear translation involves motion along straight parallel paths, like a subway train moving consistently along its track. Curvilinear translation follows curved paths that remain equidistant, similar to a roller coaster navigating winding tracks. Both preserve the body's orientation, but rectilinear paths are straight while curvilinear paths are curved.

During rotation about a fixed axis, all particles except those on the rotation axis follow circular paths around that axis. A spinning top exemplifies this motion, rotating around its central axis while remaining stationary in space. The distance from each particle to the axis remains constant throughout the rotation.

General planar motion combines both translational and rotational motion simultaneously. A frisbee demonstrates this perfectly as it spins while moving forward, keeping its rotational axis perpendicular to the plane of rotation. This complex motion integrates linear displacement with angular rotation, representing the most comprehensive type of planar rigid-body motion.

Parallel paths in rectilinear translation ensure that any two points on the rigid body move identically in direction and speed. This property means the body maintains its orientation without rotation while traveling in a straight line. The parallel nature of these paths guarantees uniform motion across all particles of the body.

In curvilinear translation, equidistant curved paths mean that any two points on the body follow identical curved trajectories separated by a constant distance. Like a snake slithering through grass, the head and tail trace the same curved path but never converge. This maintains the body's orientation while allowing smooth curved motion.

Rectilinear and curvilinear translation are both pure translational motions where the body moves without rotating. Rotation about a fixed axis involves only angular motion around a stationary point. General planar motion synthesizes these concepts by combining translation and rotation, representing the most complete form of rigid-body motion in a plane.