Overview
This lecture introduces general curvilinear motion, focusing on describing position, velocity, and acceleration of a particle moving along a curved path using vectors and rectangular (Cartesian) components.
Curvilinear Motion Fundamentals
- Curvilinear motion involves a particle moving along a curved path rather than a straight line.
- The position of a particle is described by a position vector r, measured from a fixed origin.
- The path of the particle is denoted as s, which follows the actual curved trajectory.
- Position vector r is a function of time, with both magnitude and direction that vary as the particle moves.
Displacement and Path Length
- Displacement (Δr) is the vector difference between two position vectors; it points directly from the starting to ending position.
- The change along the actual path (path length) is denoted as Δs and follows the curve.
Velocity and Speed
- Velocity is the rate of change of position with respect to time (vector quantity), always tangent to the path.
- Average velocity = Δr / Δt; Instantaneous velocity v = dr/dt.
- Speed is the magnitude (absolute value) of velocity and is not a vector; speed = |v| = ds/dt.
Rectangular (Cartesian) Components
- Motion in curvilinear paths is typically analyzed using x, y (and z in 3D) components.
Acceleration
- Acceleration is the rate of change of velocity with respect to time.
- Average acceleration = Δv / Δt; Instantaneous acceleration a = dv/dt.
- Acceleration can also be written as the second derivative of position: a = d²r/dt².
- Acceleration is always tangent to the "hodograph" (a vector diagram of velocity vectors) and, in curved motion, points toward the direction of curvature.
Key Terms & Definitions
- Position Vector (r) — A vector pointing from the origin to the particle's location.
- Displacement (Δr) — The vector change in position between two points.
- Path Length (Δs) — The distance moved along the curved path.
- Velocity (v) — The derivative of position with respect to time; tangent to the path.
- Speed — The magnitude of velocity; a scalar value.
- Acceleration (a) — The derivative of velocity with respect to time or the second derivative of position.
- Hodograph — A plot of all velocity vectors from a common origin, used to visualize changes in velocity.
Action Items / Next Steps
- Review definitions and relationships between position, velocity, speed, and acceleration.
- Prepare for introduction to projectile motion in the next session.