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