+ circular motion chin-sung lin. + rotation & revolution rotation revolution axis

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Circular Motion Chin-Sung Lin

+Rotation & Revolution

Rotation

Revolution

Axis

Axis

+Period & Frequency

Period (T): seconds/cycle

Radius (r)

Frequency (f): cycles/second (Hz)

+Period & Frequency

Radius (r)

T = 1/f

f = 1/T

+Period & Frequency Exercise

Radius (r)

If the frequency is 40 Hz, what’s the period?

+Period & Frequency Exercise

Radius (r)

If the period is 0.05 s, what’s the frequency?

+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


R

r

A A

BB

??? 2πR = 2πr ???

2πR

2πr A

B


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


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


For circular motion,

tangential speed = 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



Tangential Speed ?

4 m

Frequency = 2 Hz



Tangential Speed = 12π m/s

2 m

Frequency = ?

Period = ?



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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Radius (r)

Rotational & Linear Speed


Rotational Speed = 2π/T = 2πf (rads/s)

1 revolution = 2π

Period = T



Period = 2 s Rotational Speed = ?

5 m



Frequency = 2 Hz Rotational Speed = ?

5 m



Rotational Speed = 2πf (rads/s)

Tangential Speed v = 2πrf (m/s)

v = r

(Tangential speed) = (Radial distance) x (Rotational speed)



At the center (or axis) of the rotational platform, there is no tangential speed, but there is rotational speed



Rotational Speed = 4π

3 m

Linear Speed = ?



Linear Speed = 6π m/s

2 m

Rotational Speed = ?



Period = 3 s

4 m

Rotational Speed = ? Linear Speed = ?

2 mA

B


R

r

A

B


R

r

2πR

2πR

2πRA

B

A

B

A

B

+Centripetal Force & Acceleration


Centripetal Force

Inertia

+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


A

B

C

D

A

B

C

D

Change Speed Change Direction


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 acceleration is directed toward the center of the circle

Ac

Ac

Ac

Ac


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


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 = ?



Rotational speed = 2 rad/s

3 m

Centripetal Acceleration = ?



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 & Acceleration In linear motion

Fnet = m a In circular motion

Fc = m Ac


Fc

Fg

m

v

Ac = v 2 / r

Fc = m Ac

Fc = m v 2 / r


Fc

mv

Ac = v 2 / r

Fc = m Ac

Fc = m v 2 / r


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 (Fc):

Angular speed = 3 rad/s

2 m

Centripetal Force = ?

5 kg


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 …………


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 …………


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?


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


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?


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?


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

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Centripetal Force Lab

+Centripetal Force Lab


Fc

Fg

m

v


Fc

Fg

m

v

Ac = v 2 / r

Fc = m Ac

Fc = m v 2 / r


Common Errors

The position of clip

The plane of circular motion

The washers are not identical

+ circular motion chin-sung lin. + rotation & revolution rotation revolution axis

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