III. Kinetics of Particles

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III. Kinetics of Particles
Motion in response to non-zero forces
Combines ANALYSIS OF FORCES (as in statics, using
free body diagrams) with KINEMATICS of motion
BUT some motion is much more easily analyzed by
thinking about concepts of ENERGY and MOMENTUM
So, 3 classes of analysis
(i)
(ii) Energy Methods
(iii) Momentum methods
Remember the 5-step Program
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Example Roller coaster approaches a downhill
curve (a circular arc of radius R) at speed vo.
At what point along the curve does the car leave
the track? (Neglect friction the coaster is
rolling with minimal rolling resistance)
1. Pick coordinates
n-t
2. Draw Free Body Diagram
Gravity,
Normal
3. Set Fma
4. Solve
When does it leave track?
N0
But we dont know the velocity
Examine the other equation
This tells us acceleration versus angle (position)
Wheres the position?
By geometry, sR? !
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But we dont know the velocity
Examine the other equation
This tells us acceleration versus angle (position)
Wheres the position?
By geometry, sR? !
SO ?
Step 5 Use Kinematics
We want velocity and we have a vs. s
Separate and Integrate
Velocity as a function of angle
Normal force vs. angle
Finally, substituting in
V00 ?48.2o
Loses contact at
V0gt0 ?lt48.2o
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We can also compute the velocity vs. angle using
the work-energy approach
Only force doing work is gravity Work done by
gravity is always (with y vertical)
By geometry, height change at angle ? is
So,
Same as before, of course
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Example Coin is placed on a platform and rotated
at constant rate .
If the block slips at angle ?, what is the
static friction coefficient ?s?
1. Choose coordinates
Rotation plus possible sliding along platform
Polar r-?
2. Draw Free Body Diagram
Gravity
Normal
Friction opposes sliding downhill
3. Set Fma
4. Solve
Up until slipping,
At slipping,
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General comments
Rotation makes slipping occur at a steeper angle
Slow rotation static result ?stan? ? ?tan-1?s
Do some tests, plug in some numbers
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Example An amusement park ride has you walk into
a circular room and stand against the wall. The
room then begins to rotate. At some point, the
floor then drops, leaving you pinned against the
side wall. What is the minimum rotation rate
needed to keep you pinned? Does it depend on
your weight? Clothes you are wearing?
1. Choose coordinates
Rotational motion plus possible sliding down the
wall 3d needed
Cylindrical r-?-z
2. Draw Free Body Diagram
Gravity
Normal (wall pushing on person)
Friction opposes sliding downhill and acts to
pull person around if the ride has rotational
acceleration
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Helpful to look at top and side views of the
problem, since 3d can be hard to visualize
3. Set Fma
4. Solve
Friction in angular direction only if angular
acceleration
Friction in vertical direction to balance weight
Normal force inward due to angular velocity
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Does NOT depend on mass Does depend on R Does
depend on clothing ! Velcro is better than Lycra
Typical amusement park R10m, say ?s0.2, so
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CFD Computational Fluid Dynamics
Divide space up into a bunch of tiny blocks
Consider fluid flowing in and out of the blocks
Identify fluid behavior (viscosity ?)
Compute forces on the fluid in the block
Use Newtons laws (Fma) to determine motion of
fluid in block
Increment time by small amount ?t and update
fluid velocities
Move fluid from one block to the next using
computed velocity and time ?t