Transcript Rotational mechanics

Rotational mechanics
How can you change the rotational motion
of an object?
Take any object and apply a force
to make it spin…
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How many different ways can you make it
spin?
How many different axes of rotation can
you discover?
At what point on the object do you have to
apply this force?
Draw a 3D diagram with all the axes of rotation
Diagrams of various objects
Observation: axes of rotation intersect!
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Intersection is at geometric center for
objects that are symmetrical, same
material and solid throughout
Intersection is not at geometric center for
oddly shaped or not same material
Intersection can be in empty space,
outside the actual object: boomerang!
Observation: the axes all intersect at one point
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What does this mean? It’s the location of
the location of the ‘center of gravity’ or
‘center of mass’
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The entire weight or mass of an object can
be considered concentrated at this one
location for purposes of physics
calculations
Are CG and CM the same or not?
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CG takes into account the effects of gravity or
gravitational fields
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If a gravitational field is uniform over the whole object, ‘g’
is the same up and down, left and right, then CG and
CM are the same spot
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When might a gravitional field vary? For extremely large
objects or systems such as planetary systems
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Even for skyscrapers, CG and CM are only millimeters
apart
Other ways to find the CG?
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Try to balance the object
The location of the CG will be in a vertical
line with the balance point or over the
base of support
How did you apply the force?
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Push or pull at some point on the edge, on
one edge or both in opposite directions
force
force
Would it spin if you pushed through the CG?
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A force through the CG only affects the
linear motion
Won’t make it spin
Lever arm
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The distance between the point at which
you apply the force AND the axis of
rotation is the LEVER ARM, d
d
Axis of rotation
What is the most efficient way to
apply the force?
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Perpendicular to the lever arm
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You can apply the force at any angle you want
(with respect to the lever arm)
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In that case, the equation becomes
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Torque = force x lever arm x cosine of the angle between the
force and the lever arm
When force is perpendicular , angle is 90
degrees and cosine 90 is 1 !!
Analogy to Newton’s laws
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If force can cause a change in linear
motion, then….
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A force applied at a point other than the
CG can cause a change in ……
A change in rotational motion!!
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Torque causes a change in rotational
motion
All of Newton’s laws apply! Yay Newton!!
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If force is applied perpendicular to the
lever arm, then
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Torque
= force F x lever arm d
Since all of Newton’s laws
apply…
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If the sum of the torques = 0,
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Then the object’s rotational motion will not
change
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Wasn’t rotating….still not rotating
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(static equilibrium)
Was rotating…continues at the same rate
Since all of Newton’s laws
apply…
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If the sum of the torques is NOT = 0,
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Then the object’s rotational motion WILL
change, it will acclerate in an angular or
rotational sense
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Wasn’t rotating….starts to rotate
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Was rotating…rate of rotation is changed
Demos from class…
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If we push on the shoulders in the same
direction, do we make him spin? Why, why not?
Demos from class…
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If we push on the shoulders in the opposite
directions, do we make him spin? Why, why not?
Activity from class.....Newton’s 1st law….
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Weighing an elephant (balancing a meterstick)
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Observations of balanced system
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Heavier wts. closer to pivot pt
Lighter wts. farther away from pivot pt
Mathematical relationship
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Torque on left side = torque on right side
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Fxd = Fxd
Balanced torques…..sum of the torques = 0
Thoughts from class…
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Multiple weights? 2 on left, 1 on right??
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Add up torques
(F1 x d1) + (F2 x d2) = ( F3 x d3)
Pivot NOT
at the CG?
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Can you balance the stick if the pivot is
NOT at the CG?
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The weight acting through the CG causes
a torque on the right side
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Torque = wt x (dist between CG and pivot)
Must place a weight on left side to create
an equal and opposite torque!
Why are some objects easier to rotate than others?
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Remember resistance to motion???
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What was that called again?
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What might resistance to rotational motion
be called?
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Plain old inertia depended on…
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MASS!
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F = ma
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A lot of mass, a lot of resistance to linear motion
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Less acceleration for the same force, etc.
Demo in class….weights on stick!
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Hold stick at CG, slide wts. close to CG, try to twist or
spin the stick horizontally
CG, pivot
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Slide wts. farther out, equal distances
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Which arrangement was harder, easier to spin?
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WHY??
You changed the rotational inertia!
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Rotational inertia depends on
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Mass
Distribution of that mass
Relation to the intended axis of rotation
We could have made it even harder if we
grabbed it by the end and tried to spin it
along that axis!
Refer to book for formulas for regular shapes
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I is the letter variable for rotational inertia, aka
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Moment of inertia
Notice that for a hoop of same mass, size
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if hoop is rolled (axis of rotation is parallel to length),
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I is greater, = mr2
if hoop is spun like a top (axis of rotation is vertical ),
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I is less, = ½ mr2
Demo in class with hoop and solid disk…
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Which one rolled to bottom of ramp first?
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WHY?
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With disks of same mass and size,
disk of less I, less resistance to rolling,
accelerates down the ramp at a greater
rate, and wins the race!!
Rotational inertia and gymnastics
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You can rotate your body (in theory!)
around at least three different axes
Longitudinal (spinning like a ballerina, ice
skater)
 Transverse (flip, somersault)
 Medial (cartwheel)
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Which one presents the lowest I?
How many ways have you ever rotated
your body??
Other analogies from linear
motion?
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Linear momentum p = mass x velocity…
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Angular momentum L =
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Moment of inertia I x angular velocity
Conservation of linear momentum…
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Conservation of angular momentum
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Example in class – person spins faster on stool
when he brings arms in, thereby reducing
rotational inertia, and vice versa
Also refer to handouts from class