Physics Lesson 11 - Circular Motion Eleanor Roosevelt High School Chin-Sung Lin

Rotation & Revolution Axis Axis Rotation Revolution

Period & Frequency Frequency (f): cycles/second (Hz) 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 B 2πR B B

Rotational & Linear Speed B 2πR B B ??? 2πR = 2πr ???

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 r R

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

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

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 s Tangential Speed ? 3 m

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

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 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)

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 = 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 Linear Speed = ? 3 m

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

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

Rotational & Linear Speed B

Rotational & Linear Speed B 2πR B B 2πR

Centripetal Force & Acceleration

Centripetal Force & Acceleration Inertia

Centripetal Force & Acceleration

Centripetal Force & Acceleration

Centripetal Force & Acceleration Inertia

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

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

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 Centripetal acceleration is directed toward the center of the circle Ac

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 = v 2 / r

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

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

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

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

Centripetal Force & Acceleration 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 kg 2 m Centripetal Force = ?

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

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

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

Centripetal Force & Acceleration Centripetal force is directly proportional to linear speed squared (v2) Fc ~ v2 (Fc = m Ac = mv 2/r = mr 2)

Centripetal Force & Acceleration 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

Centripetal Force Lab

Centripetal Force Lab

Centripetal Force Lab m v Fc Fg

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

Centripetal Force Lab

Centripetal Force Lab

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