Objective Rectangular Components Normal and Tangential Components To investigate particle motion along a curved path “Curvilinear Motion” using three coordinate systems Rectangular Components Position vector r = x i + y j + z k Velocity v = vx i + vy j + vz k (tangent to path) Acceleration a = ax i + ay j +az k (tangent to hodograph) Normal and Tangential Components Position (particle itself) Velocity v = u ut (tangent to path) Acceleration (normal & tangent) Polar & Cylindrical Components

Curvilinear Motion: Cylindrical Components Section 12.8 Observed and/or guided from origin or from the center Cylindrical component Polar component “plane motion”

Application: Circular motion but observed and/or controlled from the center

Polar Coordinates Radial coordinate r Transverse coordinate q and r are perpendicular Theta q in radians 1 rad = 180o/p Direction ur and uq

Position Position vector r = r ur

Velocity Instantaneous velocity = time derivative of r Where

Velocity (con.) Magnitude of velocity Angular velocity Tangent to the path Angle = q + d d

Acceleration Instantaneous acceleration = time derivative of v

Acceleration (con.) Angular acceleration Magnitude Direction “Not tangent” Angle q + f f

Cylindrical Coordinates For spiral motion cylindrical coordinates is used r, q, and z. Position Velocity Acceleration

Time Derivative to evaluate If r = r(t) and q = q(t) If r = f(q) use chain rule

Problem The slotted fork is rotating about O at a constant rate of 3 rad/s. Determine the radial and transverse components of velocity and acceleration of the pin A at the instant q = 360o. The path is defined by the spiral groove r = (5+q/p) in., where q is in radians.

Example 12-20

Problem A collar slides along the smooth vertical spiral rod, r = (2q) m, where q is in radians. If its angular rate of rotation is constant and equal 4 rad/s, at the instant q = 90o. Determine - The collar radial and transverse component of velocity - The collar radial and transverse component of acceleration. - The magnitude of velocity and acceleration

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