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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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