Rotational speed

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Transcript Rotational speed

Chapter 8 – Rotational Motion
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Circular Motion
Rotational Inertia
Torque
Center of Mass and Center of Gravity
– Locating Center of Gravity
– Stability
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Centripetal Force
Centrifugal Force
Centrifugal Force in a Rotating Reference Frame
Simulated Gravity
Angular Momentum
Conservation of Angular Momentum
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Circular Motion
• Linear speed - the distance moved per unit time. Also called
simply speed.
• Rotational speed - the number of rotations or revolutions per
unit time.
• Rotational speed is often measured in revolutions per minute
(RPM).
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Rotation
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Circular Motion
• The linear speed is directly proportional to both rotational
speed and radial distance.
v=wr
• What are two ways that you can increase your linear speed on
a rotating platform?
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Circular Motion
• The linear speed is directly proportional to both rotational
speed and radial distance.
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v=wr
• What are two ways that you can increase your linear speed on
a rotating platform?
– Answers:
• Move away from the rotation axis.
• Have the platform spin faster.
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Example Question
You sit on a rotating platform halfway between the rotating axis and the
outer edge.
You have a rotational speed of 20 RPM and a tangential speed of 2 m/s.
What will be the linear speed of your friend who sit at the outer edge?
1) 1 m/s 2) 2 m/s 3) 3m/s 4) 4m/s
What will be his rotational speed?
1) 10 RPM 2) 20 RPM 3) 30 RPM 4) 40 RPM
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Example Question
You sit on a rotating platform halfway between the rotating axis and the
outer edge.
You have a rotational speed of 20 RPM and a tangential speed of 2 m/s.
What will be the linear speed of your friend who sit at the outer edge?
Answer: 4 m/s
What will be his rotational speed?
Answer: 20 RPM
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Rotational Inertia
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An object rotating about an axis tends to remain rotating unless
interfered with by some external influence.
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This influence is called torque.
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Rotation adds stability to linear motion.
– Examples:
• spinning football
• bicycle tires
• Frisbee
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Rotational Inertia
• The greater the distance between the bulk of an object's mass
and its axis of rotation, the greater the rotational inertia.
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Rotational Inertia
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Rotational Inertia
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Torque
• Torque is the product of the force and lever-arm distance, which
tends to produce rotation.
• Torque = force  lever arm
– Examples:
• wrenches
• see-saws
Newton’s First Law for Rotation
St = 0, no rotational motion !!!
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Class Problem
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James finds it difficult to muster
enough torque to turn the
stubborn bolt with the wrench.
He wishes he had a pipe handy
to effectively lengthen the
wrench handle, but doesn't. He
does, however, have a piece of
rope. Will torque be increased if
he pulls as hard on the rope as
shown?
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Class Problem
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No, the torque will be the same
because the lever-arm distance
is the same in both cases. The
lever arm is not the distance
between axis of turning and the
point of application of the force,
but the distance from the turning
axis to the "line of action" of the
applied force. Note the line of
action, and hence the lever arm
is the same in both cases.
The pipe that extends the length
of the wrench handle puts the
line of action farther from the
turning axis—the rope does not.
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Torque
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Center of Mass
• The center of mass of an object
is the average position of mass.
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Center of Mass
Objects tend to rotate about their center of mass.
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Class Problem
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The broom balances at its center of gravity. If you saw the broom into two
parts through the center of gravity and then weigh each part on a scale, which
part will weigh more?
1)
2)
3)
Handle side
Brush Side
Both weigh the same
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Class Problem
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The short broom part is heavier. It
balances the long handle just as
kids of unequal weights can balance
on a seesaw when the heavier kid
sits closer to the fulcrum. Both the
balanced broom and seesaw are
evidence of equal and opposite
torques—not weights.
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Class Problem
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The 40-kg woman stands at the end of a 4-meter long uniform
plank. If the maximum overhang for balance is 1 meter, estimate
the mass of the plank.
1)
2)
3)
4)
5)
10 kg
20 kg
40 kg
80 kg
160 kg
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Class Problem
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The mass of the plank is about 40kg. The plank tends to rotate like a seesaw about
a pivot point at the end of the building. Her weight multiplied by 1 meter produces a
torque that tends to rotate the system clockwise. The counterbalancing torque is
produced by the weight of the plank multiplied by the distance from the pivot point to
the plank's center of gravity. Note that this distance is also 1 meter. So both the
woman and the plank weigh the same. Their masses are equal.
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Stability
• For stability center of gravity must be over area of support.
• Examples:
• Tower of Pisa
• Touching toes with legs against wall
• Meter stick over the edge
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Angular Momentum
• Another conserved quantity is angular momentum, relating to
rotational inertia:
• Spinning wheel wants to keep on spinning, stationary wheel
wants to keep still (unless acted upon by an external rotational
force, or torque)
• Newton’s laws for linear (straight-line) motion have direct
analogs in rotational motion
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Conservation of Angular Momentum
• angular momentum = rotational inertia x rotational velocity
L=Iw
• Newton's first law for rotating systems:
– “A body will maintain its state of angular momentum unless acted
upon by an unbalanced external torque.”
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Angular Momentum
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Angular momentum is proportional to rotation speed times
rotational inertia
Rotational inertia characterized by (mass)(radius)2 distribution in
object
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Angular Momentum Conservation
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Speed up rotation by tucking in
Slow down rotation by
stretching out
Seen in diving all the time
Figure skaters demonstrate
impressively
Effect amplified by moving large
masses to vastly different radii
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Centripetal Force
• …is applied by some object.
• Centripetal means "center seeking".
Centrifugal Force
• …results from a natural tendency.
• Centrifugal means "center fleeing".
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Rotational Forces
Direction of
Motion
Centripetal
Force
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Centrifugal
Force
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What kind of motions do we feel?
• Aside from vibrations, don’t feel constant velocity
• But we can feel acceleration
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Rotating Drum Ride
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Vertical drum rotates, you’re pressed against wall
– Friction force against wall matches gravity
– Seem to stick to wall, feel very heavy
The forces real and perceived
Real Forces:
Perceived Forces:
Friction; up
Centripetal; inwards
Gravity (weight); down
Centrifugal; outwards
Gravity (weight); down
Perceived weight; down and out
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Simulated Gravity
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Clicker Problems
If you replace the wheels and tires of your car with new ones having
greater diameters, all of your speedometer readings thereafter will
A) increase
B) decrease.
C) remain the same.
Suppose the circumference of a bicycle wheel is 2 meters. If it rotates at
1 revolution per second when you are riding the bicycle, then your
speed will be
A) 3.14 m/s.
B) 2 m/s.
C) 6.28 m/s.
D) 1 m/s.
E) 3 m/s.
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Clicker Problems
An upright broom is easier to balance when the heavier
end is
A) highest, farthest from your hand.
B) nearest your hand.
C) same either way.
Put a pipe over the end of a wrench when trying to turn a
stubborn nut on a bolt, to effectively make the wrench
handle twice as long, you'll multiply the torque by
A) four.
B) eight.
C) two.
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Clicker Questions
The famous Leaning Tower of Pisa doesn't topple over because its
center of gravity is
A) displaced from its center.
B) in the same place as its center of mass.
C) stabilized by its structure.
D) relatively low for such a tall building.
E) above a place of support.
The chef at the infamous Fattening Tower of Pizza tosses a spinning
disk of uncooked pizza dough into the air. The disk's diameter
increases during the flight, while its rotational speed
A) increases.
B) decreases.
C) remains constant.
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Clicker Questions
To turn a stubborn screw, it is best to use a screwdriver that has a
A) long handle. B) wide handle. C) smooth handle. D) none of these
To kick a football so it won't topple end over end, kick it so the
force of impact extends
A) below its center of gravity.
B) through its center of gravity.
C) above its center of gravity.
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Clicker Questions
A car travels in a circle with constant speed. The net force on the
car is
A) directed forward, in the direction of travel.
B) zero because the car is not accelerating.
C) directed towards the center of the curve.
D) none of these
If the polar icecaps melted, the resulting water would spread over the
entire earth. This new mass distribution would tend to make the length
of a day
A) longer.
B) shorter at first, then longer.
C) shorter.
D) longer at first, then shorter.
E) stay the same.
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