Circular Motion Explained in Simple Terms

1. Chapter 5 Circular Motion

2. Circular Motion • Uniform Circular Motion • Radial Acceleration • Banked and Unbanked Curves • Circular Orbits • Nonuniform Circular Motion • Tangential and Angular Acceleration • Artificial Gravity Ch5b-Circular Motion-Revised 6/21/2010

3. f y  i x Angular Displacement  is the angular position. Angular displacement: Note: angles measured CW are negative and angles measured CCW are positive.  is measured in radians. 2 radians = 360 = 1 revolution Ch5b-Circular Motion-Revised 6/21/2010

4. r f y  i x Arc Length arc length = s = r  is a ratio of two lengths; it is a dimensionless ratio! Ch5b-Circular Motion-Revised 6/21/2010

5. Angular Speed The average and instantaneous angular speeds are:  is measured in rads/sec. Ch5b-Circular Motion-Revised 6/21/2010

6. Angular Velocity The average and instantaneous angular speeds are:  is actually a vector quantity. The direction of  is along the axis of rotation. Since we are concerned primarily with motion in a plane we will ignore the vector nature until we get to angular momentum when we will have to deal with it. Ch5b-Circular Motion-Revised 6/21/2010

7. f y i r  x Linear Speed An object moves along a circular path of radius r; what is its average linear speed? Note: Unfortunately the linear speed is also called the tangential speed. This can cause confusion. Also, (instantaneous values). Ch5b-Circular Motion-Revised 6/21/2010

8. Period and Frequency The time it takes to go one time around a closed path is called the period (T). Comparing to v = r: f is called the frequency, the number of revolutions (or cycles) per second. Ch5b-Circular Motion-Revised 6/21/2010

9. y v1 v1 v1 x v1 Linear and Angular Speed The linear speed depends on the radius r at which the object is located. Angular speed is independent of position. v2 r1 r2 Everyone sees the same ω, because they all experience the same rpms. Ch5b-Circular Motion-Revised 6/21/2010

10. y v v v x v Centripetal Acceleration Consider an object moving in a circular path of radius r at constant speed. Here, v  0. The direction of v is changing. If v  0, then a  0. Then there is a net force acting on the object. Ch5b-Circular Motion-Revised 6/21/2010

11. Centripetal Acceleration Conclusion: with no net force acting on the object it would travel in a straight line at constant speed It is still true that F = ma. But what acceleration do we use? Ch5b-Circular Motion-Revised 6/21/2010

12. Centripetal Acceleration The velocity of a particle is tangent to its path. For an object moving in uniform circular motion, the acceleration is radially inward. Ch5b-Circular Motion-Revised 6/21/2010

13. Centripetal Acceleration The magnitude of the radial acceleration is: Ch5b-Circular Motion-Revised 6/21/2010

14. Rotor Ride Example y fs N x w The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out. (a) What force keeps the people from falling out the bottom of the cylinder? Draw an FBD for a person with their back to the wall: It is the force of static friction. Ch5b-Circular Motion-Revised 6/21/2010

15. Rotor Ride Example (b) If s = 0.40 and the cylinder has r = 2.5 m, what is the minimum angular speed of the cylinder so that the people don’t fall out? Apply Newton’s 2nd Law: From (1) From (2): Ch5b-Circular Motion-Revised 6/21/2010

16. Unbanked Curve y N fs x w A coin is placed on a record that is rotating at 33.3 rpm. If s = 0.1, how far from the center of the record can the coin be placed without having it slip off? We’re looking for r. Draw an FBD for the coin: Apply Newton’s 2nd Law: Ch5b-Circular Motion-Revised 6/21/2010

17. Unbanked Curve From (2) What is ? Solving for r: Ch5b-Circular Motion-Revised 6/21/2010

18.  Banked Curves A highway curve has a radius of 825 m. At what angle should the road be banked so that a car traveling at 26.8 m/s has no tendency to skid sideways on the road? (Hint: No tendency to skid means the frictional force is zero.) R Take the car’s motion to be into the page. Ch5b-Circular Motion-Revised 6/21/2010

19. Banked Curves The normal force is the cause of the centripetal acceleration. Nsinθ. We need the radial position of the car itself, not the radius of the track. In either case the radius is not a uniquely determined quantity. This is the only path by which to traverse the track as there is no friction in the problem. Ch5b-Circular Motion-Revised 6/21/2010

20. Banked Curves y N  x w FBD for the car: Apply Newton’s Second Law: Ch5b-Circular Motion-Revised 6/21/2010

21. Banked Curves Rewrite (1) and (2): Divide (1) by (2): Ch5b-Circular Motion-Revised 6/21/2010

22. r Earth Circular Orbits Consider an object of mass m in a circular orbit about the Earth. v The only force on the satellite is the force of gravity. That is the cause of the centripetal force. Solve for the speed of the satellite: Ch5b-Circular Motion-Revised 6/21/2010

23. r Earth Circular Orbits Consider an object of mass m in a circular orbit about the Earth. v GMe is fixed - v is proportional to v and r are tied together. They are not independent for orbital motions. Ch5b-Circular Motion-Revised 6/21/2010

24. Circular Orbits Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours? Also need, From previous slide: Combine these expressions and solve for r: Ch5b-Circular Motion-Revised 6/21/2010

25. Circular Orbits Kepler’s Third Law It can be generalized to: Where M is the mass of the central body. For example, it would be Msun if speaking of the planets in the solar system. For non-circular orbits (elliptical) the mean radius is used and so the mean velocity is obtained. Ch5b-Circular Motion-Revised 6/21/2010

26. at a ar v Nonuniform Circular Motion Nonuniform means the speed (magnitude of velocity) is changing. There is now an acceleration tangent to the path of the particle. The net acceleration of the body is This is true but useless! Ch5b-Circular Motion-Revised 6/21/2010

27. at a ar Nonuniform Circular Motion at changes the magnitude of v. Changes energy - does work ar changes the direction of v. Doesn’t change energy - does NO WORK The accelerations are only useful when separated into perpendicular and parallel components. Can write: Ch5b-Circular Motion-Revised 6/21/2010

28. r y x N w Loop Ride Example: What is the minimum speed for the car so that it maintains contact with the loop when it is in the pictured position? FBD for the car at the top of the loop: Apply Newton’s 2nd Law: Ch5b-Circular Motion-Revised 6/21/2010

29. Loop Ride The apparent weight at the top of loop is: You lose contact when N = 0 N = 0 when This is the minimum speed needed to make it around the loop. Ch5b-Circular Motion-Revised 6/21/2010

30. y N x w Loop Ride Consider the car at the bottom of the loop; how does the apparent weight compare to the true weight? FBD for the car at the bottom of the loop: Apply Newton’s 2nd Law: Here, Ch5b-Circular Motion-Revised 6/21/2010

31. Linear and Angular Acceleration The average and instantaneous angular acceleration are:  is measured in rads/sec2. Ch5b-Circular Motion-Revised 6/21/2010

32. Linear and Angular Acceleration Recalling that the tangential velocity is vt = r means the tangential acceleration is at Ch5b-Circular Motion-Revised 6/21/2010

33. Linear and Angular Kinematics Angular Linear (Tangential) With “a”and “at” are the same thing Ch5b-Circular Motion-Revised 6/21/2010

34. Dental Drill Example A high speed dental drill is rotating at 3.14104 rads/sec. Through how many degrees does the drill rotate in 1.00 sec? Given:  = 3.14104 rads/sec; t = 1 sec;  = 0 Want . Ch5b-Circular Motion-Revised 6/21/2010

35. Dental Drill Example A high speed dental drill is rotating at 3.14104 rads/sec. What is that in rpm’s? Ch5b-Circular Motion-Revised 6/21/2010

36. v X Car Example Your car’s wheels are 65 cm in diameter and are rotating at  = 101 rads/sec. How fast in km/hour is the car traveling, assuming no slipping? Ch5b-Circular Motion-Revised 6/21/2010

37. Artificial Gravity A large rotating cylinder in deep space (g0). Ch5b-Circular Motion-Revised 6/21/2010

38. y y N x x N Artificial Gravity Top position Bottom position Apply Newton’s 2nd Law to each: N is the only force acting. It causes the centripetal acceleration. Ch5b-Circular Motion-Revised 6/21/2010

39. Space Station Example A space station is shaped like a ring and rotates to simulate gravity. If the radius of the space station is 120m, at what frequency must it rotate so that it simulates Earth’s gravity? Using the result from the previous slide: The frequency is f = (/2) = 0.045 Hz (or 2.7 rpm). Ch5b-Circular Motion-Revised 6/21/2010

40. Summary • A net force MUST act on an object that has circular motion. • Radial Acceleration ar=v2/r • Definition of Angular Quantities (, , and ) • The Angular Kinematic Equations • The Relationships Between Linear and Angular Quantities • Uniform and Nonuniform Circular Motion Ch5b-Circular Motion-Revised 6/21/2010