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King Fahd University of Petroleum

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To investigate particle motion along a curved path 'Curvilinear Motion' using ... Planer Motion. The origin point is coincide with the location of the particle. ... – PowerPoint PPT presentation

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Title: King Fahd University of Petroleum


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 vx i vy j vz k
    (tangent to path)
  • Acceleration a ax i ay j az k
    (tangent to hodograph)
  • Normal and Tangential Components
  • Polar Cylindrical Components

3
12.6 Normal and Tangential Components
  • If the path is known i.e.
  • Circular track with given radius
  • Given function
  • Method of choice in normal and tangential
    components
  • At any instant the origin is located at the
    particle it self

4
Position
  • Direct question
  • From the geometry
  • Less emphasis in n t
  • More emphasis on radius of curvature velocity and
    acceleration

5
Planer Motion
  • The origin point is coincide with the location of
    the particle.
  • 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)
  • Circle (r) radius of the circle
  • y f(x) is given by

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 12.103
  • 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 5s.

11
Problem 12.106
  • 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
Motion in a Circle
  • Recall that acceleration is defined as a change
    in velocity with respect to time.
  • Since velocity is a vector quantity, a change in
    the velocitys direction , even though the speed
    is constant, represents an acceleration.
  • This type of acceleration is known as
  • Centripetal acceleration
  • ac v2/r

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 12.106
  • 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

20
Thank You
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