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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
  • Dr. Meyassar N. Al-Haddad
  • 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
Note Put your calculator in Rad Mode
17
Thank yoy
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