Translationally accelerated Coordinate systems

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Translationally accelerated Coordinate systems
Carefully define the fixed and moving frames!
Unprimed quantities are fixed, primed are moving
So for a frame translating relative to another
(no rotation yet)
What is R0?
This automatically implies that
What happens if
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What if
How do we modify Newtons law?
Real force in the inertial reference frame (why
am I in that frame?
So if I live in the noninertial reference frame
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Purely Rotating Coordinate systems
Lets just consider rotation, assume that the
coordinate systems are otherwise identical
Unprimed quantities are fixed, primed are rotating
So the acceleration occurs solely through
rotation about some axis, n, with some angular
speed
Consider the position vector for some point in
space in the two coordinate systems
What can we say about the two position vectors?
What about the different unit vectors?
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Purely Rotating Coordinate systems
Lets just consider rotation, assume that the
coordinate systems are otherwise identical
Unprimed quantities are fixed, primed are rotating
How are the velocities related?
Why did we differentiate the primed unit vectors,
and only the primed unit vectors?
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Time-derivatives of rotating unit vectors
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Combining the last two slides for Purely Rotating
Coordinate systems
This relationship has to hold for the time
derivative of any vector! (Why?)
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Acceleration in a rotating frame
Now to find the acceleration in a rotating
coordinate frame
We have the Euler (transverse), coriolis, and
centripetal accelerations
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Acceleration in an accelerated frame
In general, we can have translation and rotation
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Forces in an accelerated frame
In an inertial frame
In a non-inertial frame
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Particle motion near the Earth
The last term is part of the measured local g, so
The transverse and centrifular forces can be
neglected (why?)
Lets pick axes so that x is east, y is north
we already said z was vertical
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Particle motion near the Earth
Lets pick axes so that x is east, y is north
we already said z was vertical
Using the rotation angle (aka latitude)
Is this a separable equa
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Particle motion near the Earth
Is this a separable equation?

• Yes
• No

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Particle motion near the Earth
Is this a separable equation?

• Yes
• No

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Particle motion near the Earth
We can integrate these equations once how can I
get away with that?
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Particle motion near the Earth
Algebra time several options I picked the easiet
I made an approximation what is it? How am I
justified in doing so?
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Particle motion near the Earth
Integrate twice
Substitute this into the equations for y and z
Now that I have a solution for particle motion
near the Earths surface, what should I do?
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Practice problems