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## King Fahd University of Petroleum

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### King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 3 12.3 Rectangular Kinematics: Erratic ... – 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
• Lecture 3

2
12.3 Rectangular Kinematics Erratic Motion
• Omitted

3
Objective
• To investigate particle motion along a curved
path using three coordinate systems
• Rectangular Components
• Normal and Tangential Components
• Polar Cylindrical Components

4
General Curvilinear Motion
• Section 12.4 in your text
• Path is described in three dimensions
• Position, velocity, and acceleration are vectors

Curvilinear motion Particle moves along a curved
path
5
Position
• S is a path function
• The position of the particle measured from a
fixed point O is given by the position vector
• r r(t)
• Example
• r sin (2t) i cos (2t) j 0.5 t k

6
Displacement
• The displacement Dr represents the change in the
particles position
• Dr r - r

7
Velocity
• Average velocity
• Instantaneous velocity
• As Dt 0 then Dr Ds
• Speed
• Since D r is tangent to the curve at P, then the
velocity is tangent to the curve

8
Acceleration
• Average acceleration
• Hodograph curve velocity arrowhead points
• Instantaneous acceleration

Hodograph
9
Acceleration (con.)
• a acts tangent to the hodograph
• a is not tangent to the path of motion
• a directed toward the inside or concave side

10
12.5 Curvilinear Motion Rectangular Components
• Rectangular x, y, z frame

11
Position
• Position vector r
• r x i y j z k
• The magnitude of r is always positive and
defined as
• Unit vector
• The direction cosines are

12
Velocity
• Velocity is the first time derivative of r
• Where
• Magnitude of velocity
• Direction is always tangent to the path

13
Problem
• The position of a particle is described by rA
2t i (t2-1) j ft. where t is in seconds.
Determine the position of the point and the speed
at 2 second.

14
Acceleration
• Acceleration is the first time derivative of v
• Where
• Magnitude of acceleration
• Direction is not tangent to the path

15
Example 12.9
• The distance of the balloon from A at 2 sec
• The magnitude and direction of velocity at 2 sec
• The magnitude and direction of acceleration at 2
sec

X8t
Position
Velocity
Acceleration
16
Example 12.10
t in second arguments in radians
At t 0.75 s find location, velocity, and
acceleration