1. 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

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

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

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

5. Position
Position vector
r = r ur

6. Velocity
Instantaneous velocity = time derivative of r
Where

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

8. Acceleration
Instantaneous acceleration = time derivative of v

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

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

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

12. 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.

13. Example 12-20

14. 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

15. Thank You