King Fahd University of Petroleum  Minerals

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Objective. To investigate particle motion along a curved path ?Curvilinear Motion" using three coordinate systemsRectangular ComponentsPosition vector r = x i y j z kVelocity v = vx i vy j vz k (tangent to path)Acceleration a = ax i ay j az k (tangent to hodo

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King Fahd University of Petroleum Minerals

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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’. un is the unit vector in normal direction ut is a unit vector in tangent direction

6. Radius of curvature (r) For the Circular motion : (r) = 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)

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 ut and un is used ub= ut x un

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