Transcript Center of Gravity

Circular Motion, Center of
Gravity, & Rotational Mechanics
Chapters 9, 10, & 11
Rotation and Revolution
Axis – straight line around which rotation
occurs
 Rotation – object turns about an internal
axis (Earth rotates around its axis)
 Revolution – when an object turns about
an external axis (Earth revolves around
the sun)
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Rotation
Revolution
Rotational Speed
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The linear speed of an object is greater near the
outer edge of a rotating object than on the inner
edge of the rotating object
Tangential Speed – the speed of something
moving along a circular path (the direction of
motion is always tangent to the circle)
Rotational Speed ( angular speed) – the
number of rotations per unit of time (expressed
in revolutions per minute or RPM)
All parts of a rotating object rotate about their
axis in the same amount of time!
Uniform Circular Motion
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Tangential speed and rotational speed are
related
Tangential Speed ~ Radial Distance x Rotational Speed
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As you move away from the center of a rotating
object, the tangential speed will increase while
your rotational speed stays the same
Centripetal Force
Centripetal Force – any force that causes
an object to follow a circular path
 When a car goes around a corner, the
friction between the tires and the road
provides the centripetal force needed to
keep the car going around the curve
 If not for the friction of the tires, the car
would continue moving in the straight-line
path
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Centripetal Force
To Have or Not to Have Centripetal
Force
Without Centripetal Force
With Centripetal Force
Centrifugal Force
Centrifugal Force – the outward force
associated with circular motion
 It is not a true force, but rather the effect
that inertia tries to place on you as you
follow a circular path
 From Newton’s First Law, the natural path
of an object is a straight-line, the
centripetal force is what keeps you going
in a circle
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Centrifugal Force
Center of Gravity
Center of Gravity – point located at the
object’s average position of weight
 For a symmetrical object, it is the
geometric center of the object
 For an irregularly shaped object, there is
more weight on end than the other, so the
center of gravity is toward the heavier end
 Objects not made of the same material
throughout (different densities), may have
the center of gravity very far from the
geometric center
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Center of Gravity
Center of Mass
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Center of Mass – the average position of all the
particles of mass that make up an object
For almost all objects on or near Earth, center of
gravity and center of mass are interchangeable
If you threw an object in the air, you’d see it
wobble around its center of gravity
The sun wobbles also!
As the planets move around the sun, they
contribute to the overall center of mass of the
solar system, so the sun wobbles off center
This is how astronomers look for planets orbiting
other stars!
Center of Mass in a Star System
Locating the Center of Gravity
The center of gravity is the balance point,
supporting that single point supports the
whole object
 If you suspend any object at a single point,
the center of gravity for that object will
hang directly below (or at) the point of
suspension
 The center of gravity may be located
where no actual material exists (i.e. a ring)
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Locating the Center of Gravity
Toppling
If the CG of an object is above the area of
support, it will remain upright
 If the CG extends outside the area of
support, the object will topple
 The Leaning Tower of Pisa does not topple
because its CG does not extend beyond
its base
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Toppling
Stability
Unstable Equilibrium – an object
balanced so that any displacement lowers
its center of gravity
 Stable Equilibrium – an object balanced
so that any displacement raises its center
of gravity (requires work)
 Neutral Equilibrium – an object balanced
to that its center of gravity is neither raised
nor lowered with displacement
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Stability
Center of Gravity of People
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When you stand upright with your arms hanging
at your sides, your CG is within your body,
typically 2 to 3 cm below your belly button
The CG is slightly lower in women than in men,
because women tend to be proportionally larger
in the pelvis and smaller in the shoulders
When you stand, your CG is somewhere above
the support base of your feet, we spread them
further apart in unstable situations (the bus)
When you bend over to touch your toes, you are
unconsciously extending the lower part of your
body, putting your CG outside of your body (so
you won’t topple over!)
Center of Gravity of People
Torque
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A torque is produced when a force is applied
with “leverage”
You use leverage when you use a screwdriver to
open a can of paint
The direction of your applied force is important,
you would never try to open a door with a
doorknob by push or pulling the doorknob
sideways
You apply your force PERPENDICULAR to the
plane of the door
Torque = force┴ x lever arm
Greater torques are produced when both the
force and lever arm are large
Torque
Torque and Center of Gravity
If the direction of force is through the CG
of the projectile, all the force can do is
move the object as a whole; there will be
no torque to turn the projectile
 If the force is directed “off center”, then in
addition to motion of the CG, the projectile
will rotate about its CG
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Rotational Inertia
An object rotating about an axis tends to
keep rotating about that axis (look
familiar?)
 Rotational Inertia – the resistance of an
object to changes in its rotational motion
 A torque is required to change the
rotational state of motion of an object
 Rotational inertia depends on the
distribution of the mass of an object
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Rotational Inertia
Rotational Inertia and Gymnastics
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Rotational inertia is least about the vertical head-to-toe
axis (longitudinal) on any person, because most of the
mass is concentrated there
A rotation of your body along this axis is easiest
The rotational inertia when your arms are extended is
3 times greater than when your arms are pulled in
You rotate about your transverse axis when you do a
flip or somersault
The rotational inertia of a gymnast is up to 20 times
greater when she is swinging in a fully extended
position from a horizontal bar than after dismount
when she somersaults in a tucked position (when she
let goes and tucks, she is automatically increasing her
rate of rotation by 20 times!)
Rotational Inertia and Gymnastics
Angular Momentum
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Angular Momentum – the “inertia of rotation” of
rotating objects
Like linear momentum, angular momentum is a vector
quantity
Rotational Velocity – when a direction is assigned to
rotational speed
Angular momentum = rotational inertia (I) x rotational velocity (ω)
Angular momentum = mass (m) x velocity (v) x radius (r)
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Newton’s 1st Law can now be restated for angular
momentum:
An object or system of objects will maintain its angular
momentum unless acted upon by an unbalanced
external torque
Angular Momentum
Conservation of Angular
Momentum
The Law of Conservation of
Momentum:
If no unbalanced external torque acts on a
rotating system, the angular momentum of
that system is constant.
 With no external torque, the product of
rotational inertia and rotational velocity at
one time will be the same at any other
time
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Conservation of Angular
Momentum
Assignment
Read Ch. 9-11 (pg. 122-164)
 Ch. 9: Do #31-38 (pg. 135), Appendix F
#1-7 (pg. 674)
 Ch. 10: Do #21-34 (pg. 148-149)
 Ch. 11: Do #33-40 (pg. 167)
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