Curvilinear Motion

1
Curvilinear Motion

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

2
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
3
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

4
Displacement

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

5
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

6
Acceleration

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

Hodograph
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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

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12.5 Curvilinear Motion Rectangular Components

• Rectangular x, y, z frame

9
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

10
Velocity

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

11
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.

12
Acceleration

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

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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
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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
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Thank yoy