Wednesday, 11/05/14 TEKS: P.4C: Analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples. By the end of today, IWBAT… Analyze characteristics of 2D circular motion. Essential Question: What are the components of circular motion? Topic: Circular Motion C-Notes!

+ Unit 4 - Circular Motion Slides with gray speckled background are extra and do not need to be incorporated into your C-Notes for today. Use them for review.

+ Rotation & Revolution Rotation Revolution 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? 1/40

+ Period & Frequency Exercise If the period is 0.05 s, what’s the frequency? Radius (r) Frequency = No. of cycles per second Therefore F=1/0.05 or F=100/5=20 Hertz

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

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

+ Rotational & Linear Speed Linear speed: distance moved per unit of time v = Δ d / Δ t 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 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 m A B

+ Rotational & Linear Speed R r A B

+ R r 2πR A B A B A B

+ Centripetal Force & Acceleration

+ Centripetal Force Inertia

+ Centripetal Force & Acceleration

+

+ Centripetal Force Inertia

Circular Motion Suppose you drive a go cart in a perfect circle at a constant speed. Even though your speed isn’t changing, you are accelerating. This is because acceleration is the rate of change of velocity (not speed), and your velocity is changing because your direction is changing! Remember, a velocity vector is always tangent to the path of motion. v v v

Tangential vs. Centripetal Acceleration 10 m/s 15 m/s So how do we calculate the centripetal acceleration ? ? ? Stay tuned! 18 m/s start finish Suppose now you drive your go cart faster and faster in a circle. Now your velocity vector changes in both magnitude and direction. If you go from start to finish in 4 s, your average tangential acceleration is: a t = (18 m/s - 10 m/s) / 4 s = 2 m/s 2 So you’re speeding up at a rate of 2 m/s per second. This is the rate at which your velocity changes tangentially. But what about the rate at which your velocity changes radially, due to its changing direction? This is your centripetal (or radial) acceleration.

+ Centripetal Force & Acceleration Centripetal Acceleration Acceleration is a vector quantity a = Δ v / Δ t Velocity can be changed by increasing/ decreasing the magnitude of v, or changing the direction

+ Centripetal Force & Acceleration Centripetal Acceleration A B C D 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 Acceleration v 0 = v r r  v f = v Let’s find a formula for centripetal acceleration by considering uniform circular motion. By the definition of acceleration, a = (v f - v 0 ) / t. We are subtracting vectors here, not speeds, otherwise a would be zero. ( v 0 and v f have the same magnitudes.) The smaller t is, the smaller  will be, and the more the blue sector will approximate a triangle. The blue “triangle” has sides r, r, and v t (from d = v t ). The vector triangle has sides v, v, and | v f - v 0 |. The two triangles are similar (side-angle-side similarity). vfvf v0v0  v f - v 0  r r v tv t continued on next slide

Centripetal Acceleration (cont.) v 0 = v r r  v f = v By similar triangles, v v  | v f - v 0 |  r r v tv t v r = v t v t So, multiplying both sides above by v, we have | v f - v 0 | t = r v 2v 2 ac =ac = Unit check: (m/s) 2 m = m2 / s2m2 / s2 m m s 2 =

Centripetal acceleration vector always points toward center of circle. v acac atat v acac atat moving counterclockwise; speeding up moving counterclockwise; slowing down “Centripetal” means “center-seeking.” The magnitude of a c depends on both v and r. However, regardless of speed or tangential acceleration, a c always points toward the center. That is, a c is always radial (along the radius).

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

+ 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 ) A c = v 2 / r

+ Centripetal Force & Acceleration Centripetal Acceleration A c = v 2 / r = (r  ) 2 / r = r  2 A c = v 2 / r = r  2

+ Centripetal Acceleration Exercise Centripetal Acceleration ( A c ): Linear speed = 6 m/s 3 m Centripetal Acceleration = ?

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

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

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

Centripetal Force, F c From F = m a, we get F c = m a c = mv 2 / r. F c = mv 2 r If a body is turning, look at all forces acting on it, and find the net force. The component of the net force that acts toward the center of curvature (perpendicular to the body’s motion) is the centripetal force. The component that acts parallel to its motion (forward or backwards) is the tangential component of the net force.

Forces that can provide a centripetal force Friction, as in the turning car example Tension, as in a rock whirling around while attached to a string, or the tension in the chains on a swing at the park.* Normal Force, as in a “round-up ride” at an amusement park (that spins & the floor drops out), or the component of normal force on a car on a banked track that acts toward the center.* Gravity: The force of gravity between the Earth and sun keeps the Earth moving in a nearly circular orbit. Any force directed toward your center of curvature, such as an applied force. * Picture on upcoming slides

+ Centripetal Force & Acceleration In linear motion F net = m a In circular motion F c = m A c

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is a force directed toward the center of the circle F c = m A c = mv 2 /r = mr  2

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

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

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to mass ( m ) F c ~ m (F c = m A c = mv 2 /r = mr  2 )

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to radius ( r ) F c ~ r (F c = m A c = mv 2 /r = mr  2 )

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

+ Centripetal Force & Acceleration Centripetal Force Centripetal force is directly proportional to angular speed squared (  2 ) F c ~  2 (F c = m A c = mv 2 /r = mr  2 )

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

+ Centripetal Force Example For a circular motion, what if mass is halved? F c will be ………… For a circular motion, what if radius is halved? F c will be ………… For a circular motion, what if linear speed is halved? F c will be ………… For a circular motion, what if angular speed is halved? F c 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? F c = 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/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 Rotation & revolution Period & frequency Linear/tangential speed: v = Δ d / Δ t = 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: A c = v 2 / r = r  2 Centripetal force: F c = m A c = mv 2 /r = mr  2 Centripetal force: F c ~ m Centripetal force: F c ~ r Centripetal force: F c ~ v 2 Centripetal force: F c ~  2