1. King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 5

2. Objective 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 = v x i + v y j + v z k (tangent to path) Acceleration a = a x i + a y j +a z k (tangent to hodograph) –Normal and Tangential Components –Polar & Cylindrical Components

3. 12.7 Normal and Tangential Components If the path is known i.e. –Circular track with given radius –Given function Method of choice is normal and tangential components

4. Position From the given geometry and/or given function More emphasis on radius of curvature velocity and acceleration

5. Planer Motion At any instant the origin is located at the particle it self The t axis is tangent to the curve at P and + in the direction of increasing s. The normal axis is perpendicular to t and directed toward the center of curvature O ’. u n is the unit vector in normal direction u t is a unit vector in tangent direction

6. Radius of curvature (  For the Circular motion : (  ) = radius of the circle For y = f(x):

7. Example Find the radius of curvature of the parabolic path in the figure at x = 150 ft.

8. Velocity The particle velocity is always tangent to the path. Magnitude of velocity is the time derivative of path function s = s(t) –From constant tangential acceleration –From time function of tangential acceleration –From acceleration as function of distance

9. Example 1 A skier travel with a constant speed of 20 ft/s along the parabolic path shown. Determine the velocity at x = 150 ft.

10. Problem A boat is traveling a long a circular curve. If its speed at t = 0 is 15 ft/s and is increasing at, determine the magnitude of its velocity at the instant t = 5 s. Note: speed increasing at # this means the tangential acceleration

11. Problem A truck is traveling a long a circular path having a radius of 50 m at a speed of 4 m/s. For a short distance from s = 0, its speed is increased by. Where s is in meters. Determine its speed when it moved s = 10 m.

12. Acceleration Acceleration is time derivative of velocity

13. Special case 1- Straight line motion 2- Constant speed curve motion (centripetal acceleration)

14. Centripetal acceleration Recall that acceleration is defined as a change in velocity with respect to time. Since velocity is a vector quantity, a change in the velocity ’ s direction, even though the speed is constant, represents an acceleration. This type of acceleration is known as Centripetal acceleration

15. Acceleration 3 types of acceleration: –linear –radial (centripetal) –angular

16. Acceleration Linear acceleration: is a change in speed without change in direction (increase in thrust in straight-and-level flight) Radial (or centripetal) acceleration : when there is a change in direction (turn, dive) Angular acceleration: when body speed and direction are changed (tight spin)

17. Problem A truck is traveling a long a circular path having a radius of 50 m at a speed of 4 m/s. For a short distance from s = 0, its speed is increased by. Where s is in meters. Determine its speed and the magnitude of its acceleration when it moved s = 10 m.

18. Review Example 12-14 Example 12-15 Example 12-16

19. Three-Dimensional Motion For spatial motion required three dimension. Binomial axis b which is perpendicular to u t and u n is used u b = u t x u n