1. King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 3

2. 12.3 Rectangular Kinematics: Erratic Motion Omitted

3. Objective To investigate particle motion along a curved path using three coordinate systems –Rectangular Components –Normal and Tangential Components –Polar & Cylindrical Components

4. Section 12.4 in your text Path is described in three dimensions Position, velocity, and acceleration are vectors

5. Position * S is a path function * The position of the particle measured from a fixed point O is given by the position vector r = r(t) Example : r = {sin (2t) i + cos (2t) j – 0.5 t k}

6. Displacement The displacement  r represents the change in the particle’s position  r = r’ - r

7. Velocity Average velocity Instantaneous velocity As  t = 0 then  r =  s Speed Since  r is tangent to the curve at P, then the velocity is tangent to the curve

8. Acceleration Average acceleration: Hodograph curve “ velocity arrowhead points ” Instantaneous acceleration: Hodograph

9. Acceleration (con.) a acts tangent to the hodograph a is not tangent to the path of motion a directed toward the inside or concave side

10. 12.5 Curvilinear Motion: Rectangular Components Rectangular : x, y, z frame

11. Position Position vector r r = x i + y j + z k The magnitude of r is always positive and defined as Unit vector The direction cosines are

12. Velocity Velocity is the first time derivative of r Where Magnitude of velocity Direction is always tangent to the path

13. Problem The position of a particle is described by r A = {2t i +(t 2 -1) j} ft. where t is in seconds. Determine the position of the point and the speed at 2 second.

14. Acceleration Acceleration is the first time derivative of v Where Magnitude of acceleration Direction is not tangent to the path

15. Example 12.9 The distance of the balloon from A at 2 sec The magnitude and direction of velocity at 2 sec The magnitude and direction of acceleration at 2 sec A Position Velocity Acceleration X=8t

16. Example 12.10 t in second arguments in radians At t = 0.75 s find location, velocity, and acceleration Note: Put your calculator in Rad Mode