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+Rotation & Revolution
Rotation
Revolution
Axis
Axis
+Period & Frequency
If the microprocessor clock of your computer is running at 2.5 GHz, what’s the period of the clock?
+Rotational & Linear Speed
R
r
A A
BB 2πR
2πr A
B
+Rotational & Linear Speed
R
r
A A
BB
??? 2πR = 2πr ???
2πR
2πr A
B
+Rotational & Linear Speed
Linear speed:
distance moved per unit of time
v = Δd / Δt
The linear speed is greater on the outer edge of a rotational object than it is closer to the axis
Rr
+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
+Rotational & Linear Speed
For circular motion,
tangential speed = linear speed
+Rotational & Linear Speed
Circumference = 2πr
Radius (r)
Linear/Tangential Speed = 2πr / T = 2πrf
Period = T
Linear / Tangential Speed (v):
+Rotational & Linear Speed Exercise
Tangential Speed ?
3 m
Period = 2 s
Linear / Tangential Speed (v):
+Rotational & Linear Speed Exercise
Tangential Speed ?
4 m
Frequency = 2 Hz
Linear / Tangential Speed (v):
+Rotational & Linear Speed Exercise
Tangential Speed = 12π m/s
2 m
Frequency = ?
Period = ?
Linear / Tangential Speed (v):
+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)
+
Radius (r)
Rotational & Linear Speed
Rotational / Angular speed ():
Rotational Speed = 2π/T = 2πf (rads/s)
1 revolution = 2π
Period = T
+Rotational & Linear Speed Exercise
Rotational / Angular speed ():
Period = 2 s Rotational Speed = ?
5 m
+Rotational & Linear Speed Exercise
Rotational / Angular speed ():
Frequency = 2 Hz Rotational Speed = ?
5 m
+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)
+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
+Rotational & Linear Speed Exercise
Rotational / Angular speed ():
Rotational Speed = 4π
3 m
Linear Speed = ?
+Rotational & Linear Speed Exercise
Rotational / Angular speed ():
Linear Speed = 6π m/s
2 m
Rotational Speed = ?
+Rotational & Linear Speed Exercise
Rotational / Angular speed ():
Period = 3 s
4 m
Rotational Speed = ? Linear Speed = ?
2 mA
B
+Rotational & Linear Speed
R
r
A
B
+Rotational & Linear Speed
R
r
2πR
2πR
2πRA
B
A
B
A
B
+Centripetal Force & AccelerationCentripetal Acceleration
Acceleration is a vector quantity
a = Δv / Δt
Velocity can be changed by increasing/ decreasing the magnitude of v, or changing the direction
+Centripetal Force & AccelerationCentripetal Acceleration
A
B
C
D
A
B
C
D
Change Speed Change Direction
+Centripetal Force & AccelerationCentripetal 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)
+Centripetal Force & AccelerationCentripetal Acceleration
Centripetal acceleration is directed toward the center of the circle
Ac
Ac
Ac
Ac
+Centripetal Force & AccelerationCentripetal 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 = v 2 / r
+Centripetal Force & AccelerationCentripetal Acceleration
Ac = v 2 / r = (r) 2 / r = r 2
Ac = v 2 / r = r 2
+Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Linear speed = 6 m/s
3 m
Centripetal Acceleration = ?
+Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Rotational speed = 2 rad/s
3 m
Centripetal Acceleration = ?
+Centripetal Acceleration Exercise
Centripetal Acceleration (Ac):
Period = 2 s Centripetal Acceleration = ?
5 m
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is a force directed toward the center of the circle
Fc
Fc
Fc
Fc
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is a force directed toward the center of the circle
Fc = m Ac = mv 2/r = mr 2
+Centripetal Force Exercise
Centripetal Force (Fc):
Linear speed = 4 m/s
2 m
Centripetal Force = ?
2 kg
+Centripetal Force Exercise
Centripetal Force (Fc):
Angular speed = 3 rad/s
2 m
Centripetal Force = ?
5 kg
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is directly proportional to mass (m)
Fc ~ m
(Fc = m Ac = mv 2/r = mr 2)
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is directly proportional to radius (r)
Fc ~ r
(Fc = m Ac = mv 2/r = mr 2)
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is directly proportional to linear speed squared (v2)
Fc ~ v2
(Fc = m Ac = mv 2/r = mr 2)
+Centripetal Force & AccelerationCentripetal Force
Centripetal force is directly proportional to angular speed squared (2)
Fc ~ 2
(Fc = m Ac = mv 2/r = mr 2)
+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 …………
For a circular motion, what if linear speed is doubled? Fc will be …………
For a circular motion, what if angular speed is doubled? Fc will be …………
+Centripetal Force Example
For a circular motion, what if mass is halved? Fc will be …………
For a circular motion, what if radius is halved? Fc will be …………
For a circular motion, what if linear speed is halved? Fc will be …………
For a circular motion, what if angular speed is halved? Fc will be …………
+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?
+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
+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?
+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?
+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?
+Summary
Rotation & revolution
Period & frequency
Linear/tangential speed: v = Δd / Δt = 2πr / T = 2πrf (m/s)
Rotational/angular speed: = 2π/T = 2πf (rads/s)
Tangential speed = Radius x Rotational speed: v = r
+Summary
Centripetal force & acceleration
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