Motion in 3 dimensions: add a third direction

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Motion in 3 dimensions add a third direction
For this course, simplest situations only add a
z coordinate to the planar coordinate systems
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Relative Motion (not your cousin moving in from
Indiana)
Motion of one body, say B, as seen by another
moving body, say A
B
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Relative Motion (continued)
A
B
O
Relative Acceleration take time derivative again
!
KEY NOTE NEWTONS 2nd LAW APPLIES TO THE TRUE
ACCELERATION (ACCELERATION MEASURED IN A
NON-ACCELERATING (INERTIAL) FRAME OF REFERENCE)
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Motion of a Water Skier Pulling boat moving at
speed of 12 m/s and acceleration of 2 m/s2 as
shown. At the instant shown, angle ?30o,
Rope length 10m

• Find the velocity of the water skier
• Find the acceleration of the water skier
• Find the radius of curvature of the skiers path

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Motion of boat is given, motion of skier relative
to the boat is given ? use relative motion
analysis
2. Coordinate system x, y shown
BUT skiers relative motion is best in POLAR
coords
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Plug in
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Radius of curvature of the path?
This is fairly tricky not circular motion since
the boat is also moving
What do we have that involves radius of curvature?
The skier is on some sort of path like this
NORMAL ACCELERATION
We have used these polar coordinates
Can now define PATH n-t coordinates
But, we dont know the normal direction..
or do we?
We have calculated the velocity
Divide acceleration into n, t components
We know velocity is in tangential direction
normal direction is perpendicular to it!
We have calculated the acceleration
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Radius of curvature of the path?
This is fairly tricky not circular motion since
the boat is also moving
What do we have that involves radius of curvature?
NORMAL ACCELERATION
But, we dont know the normal direction..or do
we?
We know velocity is in tangential direction, and
normal direction is perpendicular to tangential
direction!
Do the math on the blackboard !!
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Constrained Motion
When two bodies are tied together in some way,
the motions are not independent there is a
constraint, usually the length of cable or rod
tying the bodies together
Obviously, motions of A and B cannot be
arbitrary must stay the same
Common situation is systems of cables and pulleys
LA
LB
But total cable length is fixed ? constraint
Take time derivative of constraint equation
A gets longer, B gets shorter, and vice-versa
derived via analysis of constraint
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Example B moves down at 2 ft/s with
acceleration of 0.5 ft/s2 UP. What are
velocity and motion of A ?
1. Identify reference origins for motions
This part always has the same length
2. Measure positions relative to origins
3. Write down constraint equation
4. Take time derivative of constraint
In general Find the geometric constraints and
take time derivatives