Circular Motion ChinSung Lin Rotation Revolution Axis Rotation

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+ Circular Motion Chin-Sung Lin

+ Circular Motion Chin-Sung Lin

+ Rotation & Revolution Axis Rotation Revolution

+ Rotation & Revolution Axis Rotation Revolution

+ Period & Frequency Period (T): seconds/cycle Radius (r) Frequency (f): cycles/second (Hz)

+ 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 Radius (r) T = 1/f f = 1/T

+ Period & Frequency Exercise Radius (r) If the frequency is 40 Hz, what’s

+ 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,

+ 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

+ 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 2πr A A B 2πR B

+ Rotational & Linear Speed r R A 2πr A A B 2πR B B

+ Rotational & Linear Speed r R A 2πr A A B 2πR B

+ Rotational & Linear Speed r R A 2πr A A B 2πR B B ? ? ? 2πR = 2πr ? ? ?

+ Rotational & Linear Speed Linear speed: distance moved per unit of time r

+ Rotational & Linear Speed Linear speed: distance moved per unit of time r v = Δd / Δt The linear speed is greater on the outer edge of a rotational object than it is closer to the axis R

+ Rotational & Linear Speed Tangential speed: The speed of an object moving along

+ 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 v the circle v v v

+ Rotational & Linear Speed For circular motion, tangential speed = linear speed

+ Rotational & Linear Speed For circular motion, tangential speed = linear speed

+ Rotational & Linear Speed Linear / Tangential Speed (v): Circumference = 2πr Period

+ Rotational & Linear Speed Linear / Tangential Speed (v): Circumference = 2πr Period = T Radius (r) Linear/Tangential Speed = 2πr / T = 2πrf

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Period = 2

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Period = 2 s Tangential Speed ? 3 m

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Frequency = 2

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Frequency = 2 Hz Tangential Speed ? 4 m

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Frequency = ?

+ Rotational & Linear Speed Exercise Linear / Tangential Speed (v): Frequency = ? Tangential Speed = 12π m/s Period = ? 2 m

+ Rotational & Linear Speed Rotational / Angular speed ( ): The number of

+ 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: n Degrees/second or radians/second n Revolutions per minute (RPM)

+ Rotational & Linear Speed Rotational / Angular speed ( ): 1 revolution =

+ Rotational & Linear Speed Rotational / Angular speed ( ): 1 revolution = 2π Period = T Radius (r) Rotational Speed �= 2π/T = 2πf (rads/s)

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Period =

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Period = 2 s Rotational Speed = ? 5 m

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Frequency =

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Frequency = 2 Hz Rotational Speed = ? 5 m

+ Rotational & Linear Speed Rotational / Angular speed ( ): Rotational Speed =

+ 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

+ 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

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Rotational Speed = 4π Linear Speed = ? 3 m

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Linear Speed

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Linear Speed = 6π m/s Rotational Speed = ? 2 m

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Period =

+ Rotational & Linear Speed Exercise Rotational / Angular speed ( ): Period = 3 s 4 m 2 m Rotational Speed = ? B A Linear Speed = ?

+ Rotational & Linear Speed r R A B

+ Rotational & Linear Speed r R A B

+ Rotational & Linear Speed r R A 2πR A A B 2πR B

+ Rotational & Linear Speed r R A 2πR A A B 2πR B B 2πR

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration Centripetal Force Inertia

+ Centripetal Force & Acceleration Centripetal Force Inertia

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration

+ Centripetal Force & Acceleration Centripetal Force Inertia

+ Centripetal Force & Acceleration Centripetal Force Inertia

+ Centripetal Force & Acceleration Centripetal Acceleration is a vector quantity a = Δv

+ Centripetal Force & Acceleration Centripetal 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 & Acceleration Centripetal Acceleration A A B D C C Change

+ Centripetal Force & Acceleration Centripetal Acceleration A A B D C C Change Speed D B Change Direction

+ Centripetal Force & Acceleration Centripetal Acceleration An object moves around in a circle

+ 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)

+ Centripetal Force & Acceleration Centripetal acceleration is directed toward the center of the

+ Centripetal Force & Acceleration Centripetal acceleration is directed toward the center of the circle Ac Ac

+ Centripetal Force & Acceleration Centripetal Acceleration An acceleration that is directed at a

+ 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

+ Centripetal Force & Acceleration Centripetal Acceleration Ac = v 2 / r =

+ Centripetal Force & Acceleration Centripetal Acceleration Ac = v 2 / r = (r ) 2 / r = r 2 Ac = 2 v / r = r 2

+ Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Linear speed = 6 m/s 3 m

+ 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 Exercise Centripetal Acceleration (Ac): Rotational speed = 2 rad/s 3 m Centripetal Acceleration = ?

+ Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Period = 2 s Centripetal Acceleration =

+ Centripetal Acceleration Exercise Centripetal Acceleration (Ac): Period = 2 s Centripetal Acceleration = ? 5 m

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward the center of the circle Fc Fc

+ Centripetal Force & Acceleration In linear motion Fnet = m a In circular

+ Centripetal Force & Acceleration In linear motion Fnet = m a In circular motion Fc = m Ac

+ Centripetal Force & Acceleration m Fc v Ac = v 2 / r

+ Centripetal Force & Acceleration m Fc v Ac = v 2 / r Fc = m Ac Fc = m v 2 / r Fg

+ Centripetal Force & Acceleration v m Fc Ac = v 2 / r

+ Centripetal Force & Acceleration v m Fc Ac = v 2 / r Fc = m Ac Fc = m v 2 / r

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward the center of the circle Fc = m Ac = 2 mv /r = mr 2

+ Centripetal Force Exercise Centripetal Force (Fc): Linear speed = 4 m/s 2 m

+ Centripetal Force Exercise Centripetal Force (Fc): Linear speed = 4 m/s 2 m 2 kg Centripetal Force = ?

+ Centripetal Force Exercise Centripetal Force (Fc): Angular speed = 3 rad/s 2 m

+ Centripetal Force Exercise Centripetal Force (Fc): Angular speed = 3 rad/s 2 m 5 kg Centripetal Force = ?

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to mass

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to mass (m) Fc ~ m (Fc = m Ac = mv 2/r = mr 2)

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to radius

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to radius (r) Fc ~ r (Fc = m Ac = mv 2/r = mr 2)

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to linear

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to linear speed squared (v 2) Fc ~ 2 v (Fc = m Ac = mv 2/r = mr 2)

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to angular

+ 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)

+ Centripetal Force Example n For a circular motion, what if mass is doubled?

+ Centripetal Force Example n For a circular motion, what if mass is doubled? Fc will be ………… n For a circular motion, what if radius is doubled? Fc will be ………… n For a circular motion, what if linear speed is doubled? Fc will be ………… n For a circular motion, what if angular speed is doubled? Fc will be …………

+ Centripetal Force Example n For a circular motion, what if mass is halved?

+ Centripetal Force Example n For a circular motion, what if mass is halved? Fc will be ………… n For a circular motion, what if radius is halved? Fc will be ………… n For a circular motion, what if linear speed is halved? Fc will be ………… n 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

+ 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

+ 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 = 280 kg x (32 m/s)2/130 m = 2205 N

+ Centripetal Force Exercise Astronauts are trained to tolerate greater acceleration than the gravity

+ 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

+ 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

+ Centripetal Force Exercise At the outer edge of a rotating space station, 1 km from its center, the rotational acceleration is 10. 0 m/s 2. 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 n Rotation & revolution n Period & frequency n Linear/tangential speed: v

+ Summary n Rotation & revolution n Period & frequency n Linear/tangential speed: v = Δd / Δt = 2πr / T = 2πrf (m/s) n Rotational/angular speed: �= 2π/T = 2πf n Tangential (rads/s) speed = Radius x Rotational speed: v = r�

+ Summary n Centripetal force & acceleration n Centripetal acceleration: Ac = v 2

+ Summary n Centripetal force & acceleration n Centripetal acceleration: Ac = v 2 / r = r 2 n Centripetal force: Fc = m Ac = mv 2/r = mr 2 n Centripetal force: Fc ~ m n Centripetal force: Fc ~ r n Centripetal force: Fc ~ v 2 n Centripetal force: Fc ~ 2

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab m Fc Fg v

+ Centripetal Force Lab m Fc Fg v

+ Centripetal Force Lab m Fc v Ac = v 2 / r Fc

+ Centripetal Force Lab m Fc v Ac = v 2 / r Fc = m Ac Fc = m v 2 / r Fg

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab

+ Centripetal Force Lab Common Errors n The position of clip n The plane

+ Centripetal Force Lab Common Errors n The position of clip n The plane of circular motion n The washers are not identical