1. Circular Motion

2. Circular (Angular) Motion • Measured in degrees, and also in science, radians rather than meters as in linear motion. Otherwise, circular motion is similar to linear (translational) motion. • Radian: an angle whose arc length is equal to the radius • Since there are 2 π rad in one revolution (360˚), conversions from degrees to radians follow: • Rad = degrees x (π/180) AND • Degrees = rad x (180/ π) • 1 rad = 57.3° • 1 revolution = 2π rad

3. Angular Displacement • Defined as: the angle (in rads) through which a body is rotated in a specific direction and about a specific axis. • Variable: ø • Unit: rads • Angular displacement = change in arc length distance from axis (radius) ∆ø = ∆s r Derivations: R = ∆s and ∆s = ∆ø(r) ∆ø

4. Example: • If the arc length of a wheel is .50 meters, and the radius is 0.23 meters, what is the angular displacement of an ant riding along ? ∆ø = ∆s r ∆ø = .50 0.23 ∆ø = 2.17 rads

5. Angular Velocity • Defined as: A measure of the rate of change in angular position • Variable: • Unit: radians per second • = ∆ø ∆T This is the same as : v = 2πr= linear or tangential speed ∆T and if Given mass & Centripetal Force; V= √Fcr/m

6. Example • What is the angular velocity of a car on a ferris wheel if the car moves a displacement of 15 rads in 22.5 seconds? • = ∆ø ∆T = 15 22.5 = 0.67 rads/sec

7. Angular Acceleration • Occurs when there is a change in angular velocity • Variable: • = ∆÷ ∆ t • Unit: rad/s2

8. Example • a car on the ferris wheel starts at rest and has a velocity of 10 rads/s at top speed, what is the angular acceleration after 60 seconds? • = ∆÷ ∆ t • = 10 – 0 60 = 0.16 rads/s2

9. Centripetal Motion • Remember: Newton's second law of motion said the direction of the net force is in the same direction as the acceleration… SO… for an object moving in a circle, there must be an inward force (perpendicular to the direction of velocity) acting upon it in order to cause its inward acceleration  

10. Centripetal Motion • “Center Seeking” • some physical force pushing or pulling an object towards the center of the circle • This force can accelerate the object - by changing its direction - but it cannot change its speed. to change the speed , there would have to be a force in the direction of the motion of the object.

11. Centripetal Force & Acceleration • Fc = mv2 r Ac = v2 r

12. Example: • A race car is travelling around a circular track and is 20 meters from the center of the track. If his velocity is 8.5 m/s, what is his centripetal acceleration? Ac = v2 r Ac = 8.52 20 Ac = 3.6 m/s2

13. Centripetal ForceEX: As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

14. Centripetal Force • EX: As a car makes a turn, the force of friction acting upon the turned wheels of the car provides centripetal force required for circular motion.

15. Centripetal Force & Acceleration • Period: time it takes for one full revolution of an object. Measured in seconds (s) • Frequency: number of revolutions per unit of time. Measured in Hertz (Hz) • T = 1/F AND F = 1/T

16. Centrifugal Force • Newton’s third law tells us there is an equal & opposite force paired with every force… • For Centripetal Force, this is an equal force which pulls outward during circular motion, called centrifugal Force… • However, this is merely the inertia of an object resisting the changes of direction, as it tends to continue in a straight line motion