Download Centripetal Force

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Circular Motion
Chin-Sung Lin
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Rotation & Revolution
Axis
Axis
Rotation
Revolution
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Period & Frequency
Period (T): seconds/cycle
Radius (r)
Frequency (f): cycles/second (Hz)
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Period & Frequency
Radius (r)
T = 1/f
f = 1/T
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Period & Frequency Exercise
Radius (r)
If the frequency is 40 Hz, what’s the period?
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Period & Frequency Exercise
Radius (r)
If the period is 0.05 s, what’s the frequency?
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Period & Frequency
If the microprocessor clock of your computer is
running at 2.5 GHz, what’s the period of the clock?
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Rotational & Linear Speed
r
R
A
2πr
A
A
B
2πR
B
B
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Rotational & Linear Speed
r
R
A
2πr
A
A
B
2πR
B
B
??? 2πR = 2πr ???
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Rotational & Linear Speed
Linear speed:
distance moved per unit of time
r
R
v = Δd / Δt
The linear speed is greater on the outer edge
of a rotational object than it is closer to the axis
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Rotational & Linear Speed
Tangential speed:
The speed of an object
moving along a circular path
can be called tangential
speed because the direction
of motion is always tangent
to the circle
v
v
v
v
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Rotational & Linear Speed
For circular motion,
tangential speed = linear speed
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Rotational & Linear Speed
Linear / Tangential Speed (v):
Circumference = 2πr
Period = T
Radius (r)
Linear/Tangential Speed = 2πr / T = 2πrf
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Rotational & Linear Speed Exercise
Linear / Tangential Speed (v):
Period = 2 s
Tangential Speed ?
3m
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Rotational & Linear Speed Exercise
Linear / Tangential Speed (v):
Frequency = 2 Hz
Tangential Speed ?
4m
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Rotational & Linear Speed Exercise
Linear / Tangential Speed (v):
Frequency = ?
Tangential Speed = 12π m/s
Period = ?
2m
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Rotational & Linear Speed
Rotational / Angular speed ( ):
The number of rotations per unit of time
All parts of a rotational object have the same rate
of rotation, or same number of rotations per unit of
time
Unit of rotational speed:
 Degrees/second or radians/second
 Revolutions per minute (RPM)
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Rotational & Linear Speed
Rotational / Angular speed ( ):
1 revolution = 2π
Period = T
Rotational Speed
Radius (r)
= 2π/T = 2πf
(rads/s)
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Rotational & Linear Speed Exercise
Rotational / Angular speed ( ):
Period = 2 s
Rotational Speed = ?
5m
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Rotational & Linear Speed Exercise
Rotational / Angular speed ( ):
Frequency = 2 Hz
Rotational Speed = ?
5m
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Rotational & Linear Speed
Rotational / Angular speed ( ):
Rotational Speed  = 2πf (rads/s)
Tangential Speed v = 2πrf (m/s)
v = r
(Tangential speed) = (Radial distance) x (Rotational speed)
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Rotational & Linear Speed
Rotational / Angular speed ( ):
At the center (or axis) of the rotational platform,
there is no tangential speed, but there is rotational
speed
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Rotational & Linear Speed Exercise
Rotational / Angular speed ( ):
Rotational Speed = 4π
Linear Speed = ?
3m
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Rotational & Linear Speed Exercise
Rotational / Angular speed ( ):
Linear Speed = 6π m/s
Rotational Speed = ?
2m
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Rotational & Linear Speed Exercise
Rotational / Angular speed ( ):
Period = 3 s
4m
A
2m
Rotational Speed = ?
B
Linear Speed = ?
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Rotational & Linear Speed
r
R
A
B
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Rotational & Linear Speed
r
R
A
2πR
A
A
B
2πR
B
B
2πR
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Centripetal Force & Acceleration
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Centripetal Force & Acceleration
Centripetal Force
Inertia
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Centripetal Force & Acceleration
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Centripetal Force & Acceleration
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Centripetal Force & Acceleration
Centripetal Force
Inertia
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Centripetal Force & Acceleration
Centripetal Acceleration
Acceleration is a vector quantity
a = Δv / Δt
Velocity can be changed by increasing/
decreasing the magnitude of v, or changing
the direction
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Centripetal Force & Acceleration
Centripetal Acceleration
A
A
B
D
C
C
Change Speed
D
B
Change Direction
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Centripetal Force & Acceleration
Centripetal Acceleration
An object moves around in a circle with
constant speed has acceleration, because
its direction is constantly changing
This acceleration is called centripetal
acceleration (Ac)
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Centripetal Force & Acceleration
Centripetal Acceleration
Centripetal acceleration is directed toward
the center of the circle
Ac
Ac
Ac
Ac
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Centripetal Force & Acceleration
Centripetal Acceleration
An acceleration that is directed at a right
angle to the path of a moving object and
produces circular motion
Centripetal acceleration (Ac)
Ac =
2
v
/r
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Centripetal Force & Acceleration
Centripetal Acceleration
Ac = v 2 / r = (r) 2 / r = r 2
Ac =
2
v
/r=
2
r
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Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Linear speed = 6 m/s
3m
Centripetal Acceleration = ?
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Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Rotational speed = 2 rad/s
3m
Centripetal Acceleration = ?
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Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Period = 2 s
Centripetal Acceleration = ?
5m
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is a force directed toward
the center of the circle
Fc
Fc
Fc
Fc
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Centripetal Force & Acceleration
In linear motion
Fnet = m a
In circular motion
Fc = m Ac
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Centripetal Force & Acceleration
m
Fc
v
Ac = v 2 / r
Fc = m Ac
Fc = m v 2 / r
Fg
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Centripetal Force & Acceleration
v
m
Fc
Ac = v 2 / r
Fc = m Ac
Fc = m v 2 / r
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is a force directed
toward the center of the circle
Fc = m A c =
2
mv /r
=
2
mr
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Centripetal Force Exercise
Centripetal Force (Fc):
Linear speed = 4 m/s
2m
2 kg
Centripetal Force = ?
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Centripetal Force Exercise
Centripetal Force (Fc):
Angular speed = 3 rad/s
2m
5 kg
Centripetal Force = ?
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is directly proportional
to mass (m)
Fc ~ m
(Fc = m Ac = mv 2/r = mr 2)
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is directly proportional
to radius (r)
Fc ~ r
(Fc = m Ac = mv 2/r = mr 2)
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is directly proportional
to linear speed squared (v2)
Fc ~
2
v
(Fc = m Ac = mv 2/r = mr 2)
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Centripetal Force & Acceleration
Centripetal Force
Centripetal force is directly proportional
to angular speed squared (2)
Fc ~
2
(Fc = m Ac = mv 2/r = mr 2)
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Centripetal Force Example

For a circular motion, what if mass is
doubled? Fc will be …………

For a circular motion, what if radius is
doubled? Fc will be …………
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For a circular motion, what if linear
speed is doubled? Fc will be …………
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For a circular motion, what if angular
speed is doubled? Fc will be …………
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Centripetal Force Example
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For a circular motion, what if mass is
halved? Fc will be …………
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For a circular motion, what if radius is
halved? Fc will be …………
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For a circular motion, what if linear
speed is halved? Fc will be …………
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For a circular motion, what if angular
speed is halved? Fc will be …………
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Centripetal Force Example
A 280 kg motorcycle traveling at 32
m/s enters a curve of radius = 130 m.
What force of friction is required from
the contact of the tires with the road to
prevent a skid?
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Centripetal Force Example
A 280 kg motorcycle traveling at 32
m/s enters a curve of radius = 130 m.
What force of friction is required from
the contact of the tires with the road to
prevent a skid?
Fc = 280kg x (32 m/s)2/130m = 2205 N
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Centripetal Force Exercise
Astronauts are trained to tolerate
greater acceleration than the gravity
by using a spinning device whose
radius is 10.0 m. With what linear
speed and rotational speed would an
astronaut have to spin in order to
experience an acceleration of 3 g’s at
the edge of the device?
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Centripetal Force Exercise
To swing a pail of water around in a
vertical circle fast enough so that the
water doesn’t spill out when the pail is
upside down. If Mr. Lin’s arm is 0.60 m
long, what is the minimum speed with
which he can swing the pail so that
the water doesn’t spill out at the top of
the path?
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Centripetal Force Exercise
At the outer edge of a rotating space
station, 1 km from its center, the
rotational acceleration is 10.0 m/s2.
What is the new weight of a 1000 N
object being moved to a new storage
room which is 500 m from the center
of the space station?
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Summary
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Rotation & revolution
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Period & frequency
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Linear/tangential speed: v = Δd / Δt = 2πr / T = 2πrf (m/s)
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Rotational/angular speed:
 Tangential
= 2π/T = 2πf
(rads/s)
speed = Radius x Rotational speed: v = r
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Summary
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Centripetal force & acceleration
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Centripetal acceleration: Ac = v 2 / r = r 2

Centripetal force: Fc = m Ac = mv 2/r = mr 2

Centripetal force: Fc ~ m

Centripetal force: Fc ~ r

Centripetal force: Fc ~ v2

Centripetal force: Fc ~ 2
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Centripetal Force Lab
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Centripetal Force Lab
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Centripetal Force Lab
m
Fc
Fg
v
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Centripetal Force Lab
m
Fc
v
Ac = v 2 / r
Fc = m Ac
Fc = m v 2 / r
Fg
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Centripetal Force Lab
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Centripetal Force Lab
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Centripetal Force Lab
Common Errors

The position of clip
 The
plane of circular motion
 The
washers are not identical