physicsclassroom/mmedia/circmot/circmotTOC.html

1. CIRCULAR MOTION http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html

2. Uniform Circular Motion • motion of an object in a circle with a constant or uniform speed • constant change in direction The direction of the velocity vector at every instant is in the direction tangent to the circle

3. Uniform Circular Motion: Period What is this uniform speed? The speed of the object is the distance it covers ( the circumference of the circle, 2лr) divided by time T (period or the to make one complete revolution

4. Are all people on Earth moving at the same speed? • Earth is rotating about an axis through its poles • So that means we are all moving since we are all on the Earth. • Are some of us moving with a greater LINEAR SPEED than others?? • Yes, closer to the Equator, the faster you are moving…. Closer to poles, the slower you are moving

5. CENTRIPETAL ACCELERATION dh vh A Equation 1 dh = vht h = dv r r2 Equation 2 dv = ½act² Equation 3 dh² = 2rdv The speed is constant but the direction is changing with time

6. Centripetal Acceleration Substitute Eq. 1 & Eq. 2 into Eq. 3 Equation 4. ac = vh² r Sincevh = 2лr/T, then Equation 5. ac = 4л²r T² If T= 1/f, then Equation 6. ac = 4л²rf²

7. Centripetal Force CENTRIPETAL FORCE Always points toward center of circle. (Always changing direction!) Centripetal force is the magnitude of the force required to maintain uniform circular motion.

8. Centripetal means “center- seeking”. The force pushes you toward the center of the circle and keeps you moving in a circle. It keeps your inertia from taking you in a straight line. Centripetal Force is affected by mass (m), linear speed(vt), and radius (r)

9. Direction of Centripetal Force, Acceleration and Velocity Without a centripetal force, an object in motion continues along a straight-line path.

10. Direction of Centripetal Force, Acceleration and Velocity

11. 1. When driving in a circle, in what direction is a force acting on you? Pushing you outward from the circle, or inward? Ans. -- Inwards, toward the center of the circle 2. If you are swinging a yo-yo in a circle, and the string breaks…. What path does the yo – yo take? Ans -- yo- yo goes in a path tangent to the circle HOWEVER, People commonly think there is a force pushing you out from the circle; Feels like you are being pushed outward Example : The Rotor- amusement park ride, a centrifuge, CD on your dashboard moving to the right when your turning left . Why is this?

12. What if velocity decreases?

13. What if mass decreases?

14. What if radius decreases?

15. What provides the centripetal force? Tension Gravity Friction Normal Force Centripetal force is NOT a fundamental force. Acceleration is the result of the net force and centripetal force is derived only from the Newton’s 2nd law. So centripetal force depends only on the net force applied to a body moving in uniform circular motion

16. Tension Can Yield a Centripetal Acceleration If the person doubles the speed of the airplane, what happens to the tension in the cable? Doubling the speed, quadruples the force (tension) required to keep the plane in uniform circular motion.

17. Friction Can Yield a Centripetal Acceleration:

19. Example: A car traveling around a circular track Fc = Ff mv²/r = µFn µ = v² rg Frictional force between the tire and the road provides the centripetal acceleration. For the car to make the turn without skidding, a minimum coefficient of static friction must be present between the tire and the road

20. Sample problem A car is traveling at 9 m/s in a circle that has a radius of 60 m. What must be the minimum value of µ for the car to make turn without skidding? Given: v= 9m/s r= 60 m Find: µ µ = v²/rg Solution: µ = v²/rg = (9m/s)²/ (60m)(9.8 m/s²) Answer: µ = 0.14

21. Gravity Can Yield a Centripetal Acceleration

22. Gravity and Centripetal Acceleration Centripetal acceleration provided by gravitational force

23. Hubble Space Telescope orbits at an altitude of 598 km (height above Earth’s surface). What is its orbital speed?

24. The Normal Force Can Yield a Centripetal Acceleration: Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

25. Banked Curves Q: Why exit ramps in highways are banked? Answer: To increase the centripetal force for the higher exit speed.

26. Banked Curves r FNcosq = mg Fc = FN sinq = mv² r

27. Fw=FN cosq mg = FN cosq Fc =FN sinq mv²/r = FN sinq Fc = mv²/r = FN sinq Fw mg FN cosq v²/rg = tan q q = tan-1 v²/rg

28. The Normal Force and Centripetal Acceleration: How to bank a curve… …so that you don’t rely on friction at all

29. Sample Problem: A car is to make turn with a radius of curvature of 60 m at a speed of 27 m/s. at what angle should the road be banked for the car to make the turn? Given: v= 27 m/s r= 60 m Find : Ɵ Ɵ = tan-1 v²/rg Solution: Ɵ = tan-1 v²/rg Ɵ = tan-1 (27m/s) ² (60 m) (9.8m/s²) Ɵ = 51º

30. Vertical Circular Motion

31. Artifical Gravity

32. Sample Problem A rotating room has a radius of 4.5 m and the speed of the rider is 12 m/s. how much should the coefficient of static friction be to keep the rider pinned against the wall? Given: r = 4.5 m v= 12/ms Find µ µ = rg/v² Solution: µ = rg/v² µ = (4.5m)(9.8 m/s²)/ (12 m/s) ² µ = 0.31

33. The Rotor People stand with backs against wall of a large cylinder, cylinder then starts spinning, and people are seemingly pushed against the wall, then floor drops, and people are stuck against the wall.

36. So why is there no Force pushing you out from the circle? A force does not cause this…… your INERTIA does! Inertia makes you want to stay in a straight line, and by going in a circle, you are fighting your own inertia Ex: This is how Rotor works, and why CD on dashboard happens, The only actual force acting on you is the Centripetal Force

37. “Centrifugal Force • “centrifugal force” is a fictitious force - it is not an interaction between 2 objects, and therefore not a real force.Nothing pulls an object away from the center of the circle.

38. “Centrifugal Force” What is erroneously attributed to “centrifugal force” is actually the action of the object’s inertia - whatever velocity it has (speed + direction) it wants to keep.

39. Relationship Between Variables of Uniform Circular Motion • Suppose two identical objects go around in horizontal circles of identical diameter but one object goes around the circle twice as fast as the other. The force required to keep the faster object on the circular path is • the same as • one fourth of • half of • twice • four times • the force required to keep the slower object on the path. The answer is E. As the velocity increases the centripetal force required to maintain the circle increases as the square of the speed.

40. Relationship Between Variables of Uniform Circular Motion Suppose two identical objects go around in horizontal circles with the same speed. The diameter of one circle is half of the diameter of the other. The force required to keep the object on the smaller circular path is the same as • one fourth of • half of • twice • four times the force required to keep the object on the larger path. The answer is D. The centripetal force needed to maintain the circular motion of an object is inversely proportional to the radius of the circle. Everybody knows that it is harder to navigate a sharp turn than a wide turn.

41. Relationship Between Variables of Uniform Circular Motion Suppose two identical objects go around in horizontal circles of identical diameter and speed but one object has twice the mass of the other. The force required to keep the more massive object on the circular path is • the same as • one fourth of • half of • twice • four times Answer: D.The mass is directly proportional to centripetal force.

42. Centripetal Force: Question A car travels at a constant speed around two curves. Where is the car most likely to skid? Why? Smaller radius: larger force required to keep it in uniform circular motion.

43. The End!