Path or NormalTangential Coordinates:

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Path or Normal-Tangential Coordinates
When the path is fixed in space, we can use the
geometry of the path to define orthogonal (right
angle) coordinates
Points IN toward curvature
As the body moves along the path, the unit
vectors change
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• Radius of Curvature
• At any point on a curve, you can think about a
  circle that is tangent to the curve at that
  point.

• There is one circle that has the same curvature
  as the curve at that point
• The radius of that circle is the radius of
  curvature

What is ? for a straight line?
Given a path in x-y space described as yf(x),
the radius of curvature is
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What about the Motion Velocity and Acceleration ?
Velocity must always be tangent to the path. If
it werent, the body would move OFF the path, and
we have specified that it move ON the path.
Acceleration For a split instant of time, the
body seems to be moving on a circle of radius
? To curve with the path, the body must be
accelerating inward in addition to speeding up or
slowing down
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To stay ON the path, it must move inward a
distance ?s during the time ?t
For small ?t, we have ?sltlt?, and so
Distance moved by a particle under constant
acceleration a, starting from rest at time t0
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1. Specified path use n-t coordinates
Force on person in the car
Total upward force Fseat mg ma 0.8mg
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What is condition for going airborne?
Normal force 0 (no contact with ground)
Highway design better not have 300ft curvature!!
See the video on Airborne Cars.
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Race Walking same problem. How fast can a
person walk (walk one foot on the ground at all
times)?
L
Torso moving on a path with curvature leg
length L
Look at forces on torso at the peak of the curve
Gravity and normal force
Use n-t coordinates, ?L
What is condition for feet off the ground?
Normal force 0 (no contact with ground)
Max walking velocity
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Polar Coordinates
Position measured by distance r from the origin
and angle ? measured with respect to a fixed line
Unit vectors CHANGE as body moves
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Look carefully at the geometry changes as the
body moves
As for n-t coordinates, the unit vectors also
change with time
Change is in the -er direction
Change is in the e? direction
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Acceleration in Polar Coordinates Use
fundamental definition, chain rule, and results
from last slide..
Acceleration in radial direction
Acceleration in angular direction
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Polar Coordinates Summary of Results
Never need to re-derive just understand origin
and notation !!