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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 = v x i + v y j + v z k (tangent to path)

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Objective


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Presentation Transcript
slide1

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
      • Position (particle itself)
      • Velocity v = u ut (tangent to path)
      • Acceleration (normal & tangent)
    • Polar & Cylindrical Components
curvilinear motion cylindrical components
Curvilinear Motion: Cylindrical Components
  • Section 12.8
  • Observed and/or guided from origin or from the center
  • Cylindrical component
  • Polar component “plane motion”
polar coordinates
Polar Coordinates
  • Radial coordinate r
  • Transverse coordinate
  • q and r are perpendicular
  • Theta q in radians
  • 1 rad = 180o/p
  • Direction ur and uq
position
Position
  • Position vector
  • r = r ur
velocity
Velocity
  • Instantaneous velocity = time derivative of r
  • Where
velocity con
Velocity (con.)
  • Magnitude of velocity
  • Angular velocity
  • Tangent to the path
  • Angle = q + d

d

acceleration
Acceleration
  • Instantaneous acceleration = time derivative of v
acceleration con
Acceleration (con.)
  • Angular acceleration
  • Magnitude
  • Direction “Not tangent”
  • Angle q + f

f

cylindrical coordinates
Cylindrical Coordinates
  • For spiral motion cylindrical coordinates is used r, q, and z.
  • Position
  • Velocity
  • Acceleration
slide12

Problem

  • The slotted fork is rotating about O at a constant rate of 3 rad/s. Determine the radial and transverse components of velocity and acceleration of the pin A at the instant q = 360o. The path is defined by the spiral groove r = (5+q/p) in., where q is in radians.
slide14

Problem

A collar slides along the smooth vertical spiral rod, r = (2q) m, where q is in radians. If its angular rate of rotation is constant and equal 4 rad/s, at the instant q = 90o. Determine

- The collar radial and transverse component of velocity

- The collar radial and transverse component of acceleration.

- The magnitude of velocity and acceleration