1. ENGR 215 ~ Dynamics Sections 12.7

2. Lecture Example 1: Curvilinear motion Consider the function What is the acceleration along the path of travel at time, t=1sec?

4. At time, t=1 sec.

6. Dot Product

7. Tangential Acceleration

8. Cross Product Magnitude Direction

9. Calculating a Cross Product

10. Centripetal Acceleration

11. Acceleration along a path

12. Derivation of the normal and tangential components of acceleration.

15. Putting it all together…

16. From the calculus book…curvature

17. Normal and Tangential Directions n – direction points to center of arc. t x n = b

18. 3-D Motion

19. Lecture Example 2: Find the equation of the path, y=f(x). Find the normal and tangential component of the acceleration at t=0.25s.

20. Lecture Example 3: At time, t = π seconds determine the velocity of the particle, and determine the tangential and normal components of the acceleration. Draw and label the acceleration vectors on the graph below. The position of the particle is expressed in meters. Trigonometric functions should be evaluated in radians. r(t) = (t sin t) i + (t cos t ) j 0 < t < 2 π sec Spiral of Archimedes

21. Lecture Example 4: The jet plane travels along the vertical parabolic path. When at Point A it has reached a speed of 200 m/s which is increasing at a rate of 0.8 m/s 2. Determine the magnitude of the acceleration of the plane when it is at Point A. Note: The positions x and y are given in kilometers.