Rotating Coordinate Systems
based on FW-6,7,8
Sometimes it is useful to analyze motion in a non inertial reference frame, e.g.
when the observer is moving (accelerating).
Inertial frame:
orthonormal coordinate
system (fixed)
Rotating frame
(“body fixed”):
fixed
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Infinitesimal rotations:
we need to figure out how the body-fixed vectors change:
(as seen from the inertial frame)
can be expanded in
the original basis
we only need three components:
similarly for the other two:
infinitesimal rotations about
each coordinate axes
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Example:
rotation about the 1st axis:
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Change in an element of time:
instantaneous angular velocity vector of the
rotating frame as seen in the inertial frame
now we know how the body-fixed vectors change:
we get an important formula:
valid for any vector!
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operator equation, valid for any vector!
Velocity:
Rate of change of angular velocity:
Acceleration:
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Adding translations and rotations:
if the origin of the body fixed frame is also moving with respect to the origin of
the inertial reference frame:
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Newton’s Laws in Accelerated Frames
based on FW-10
Newton’s second law is valid in the inertial reference frame:
and is experienced by an observer in accelerated frame as:
force from the acceleration of the
origin of the body system
Coriolis force
force from the
angular acceleration
centrifugal force
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Motion on the Surface of the Earth
.
based on FW-11
is a simple example of what we have discussed.
m
center of the Sun
(inertial frame)
external forces
(gravity from the
Sun and from the
Earth, and others)
angular frequency from
the circular trajectory
about the Sun and from
daily rotation
(approximately constant)
center of the Earth
(rotating frame)
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rotation of the Earth:
gravity from Earth/Sun:
circular trajectory about the Sun:
Sun’s gravity cancels the
second term at the center
of the earth
approximate equation of motion for a particle on the rotating earth:
other forces
on earth
Coriolis force
centrifugal force
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Particle (stationary) on a scale:
at the poles
at the equator
in spherical polar coordinates:
net tangential component points toward the equator
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