1. Chapter 5 Circular Motion

2. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 2 Circular Motion Uniform Circular Motion Radial Acceleration Banked and Unbanked Curves Circular Orbits Nonuniform Circular Motion Tangential and Angular Acceleration Artificial Gravity

3. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 3 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 x y ii ff 

4. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 4 x y ii ff  r arc length = s = r   is a ratio of two lengths; it is a dimensionless ratio! Arc Length

5. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 5 The average and instantaneous angular speeds are:  is measured in rads/sec. Angular Speed

6. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 6 The average and instantaneous angular speeds are: Angular Velocity 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.  is actually a vector quantity.

7. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 7 An object moves along a circular path of radius r; what is its average linear speed? Also, (instantaneous values). x y ii ff r  Linear Speed Note: Unfortunately the linear speed is also called the tangential speed. This can cause confusion.

8. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 8 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. Period and Frequency

9. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 9 Angular speed is independent of position. The linear speed depends on the radius r at which the object is located. x y v1v1 v1v1 v1v1 v1v1 Linear and Angular Speed r1r1 r2r2 v2v2 Everyone sees the same ω, because they all experience the same rpms.

10. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 10 Centripetal Acceleration 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. Consider an object moving in a circular path of radius r at constant speed. x y v v v v

11. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 11 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? Centripetal Acceleration

12. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 12 The velocity of a particle is tangent to its path. For an object moving in uniform circular motion, the acceleration is radially inward. Centripetal Acceleration

13. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 13 The magnitude of the radial acceleration is: Centripetal Acceleration

14. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 14 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: x y w N fsfs It is the force of static friction. Rotor Ride Example

15. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 15 (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 2 nd Law: From (2): From (1) Rotor Ride Example

16. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 16 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. Apply Newton’s 2 nd Law: x y w N fsfs Draw an FBD for the coin: Unbanked Curve

17. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 17 From (2) Solving for r: What is  ? Unbanked Curve

18. MFMcGraw-PHY 1401Ch5b-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.)  Take the car’s motion to be into the page. R

19. MFMcGraw-PHY 1401Ch5b-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.

20. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 20 FBD for the car: x y N w  Apply Newton’s Second Law: Banked Curves

21. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 21 Rewrite (1) and (2): Divide (1) by (2): Banked Curves

22. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 22 Circular Orbits Consider an object of mass m in a circular orbit about the Earth. Earth r 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: v

23. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 23 Circular Orbits Consider an object of mass m in a circular orbit about the Earth. Earth r GM e is fixed - v is proportional to v v and r are tied together. They are not independent for orbital motions.

24. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 24 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? From previous slide: Also need, Combine these expressions and solve for r: Circular Orbits

25. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 25 Kepler’s Third Law It can be generalized to: Where M is the mass of the central body. For example, it would be M sun 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. Circular Orbits

26. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 26 Nonuniform Circular Motion There is now an acceleration tangent to the path of the particle. The net acceleration of the body is atat v arar a Nonuniform means the speed (magnitude of velocity) is changing. This is true but useless!

27. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 27 atat arar a a t changes the magnitude of v. Changes energy - does work a r changes the direction of v. Doesn’t change energy - does NO WORK Can write: Nonuniform Circular Motion The accelerations are only useful when separated into perpendicular and parallel components.

28. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 28 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: Nw y x Apply Newton’s 2 nd Law: r Loop Ride

29. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 29 The apparent weight at the top of loop is: N = 0 when This is the minimum speed needed to make it around the loop. Loop Ride You lose contact when N = 0

30. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 30 Consider the car at the bottom of the loop; how does the apparent weight compare to the true weight? N w y x FBD for the car at the bottom of the loop: Apply Newton’s 2 nd Law: Here, Loop Ride

31. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 31 Linear and Angular Acceleration The average and instantaneous angular acceleration are:  is measured in rads/sec 2.

32. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 32 Recalling that the tangential velocity is v t = r  means the tangential acceleration is atat Linear and Angular Acceleration

33. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 33 Linear (Tangential) Angular With Linear and Angular Kinematics “a” and “a t ” are the same thing

34. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 34 A high speed dental drill is rotating at 3.14  10 4 rads/sec. Through how many degrees does the drill rotate in 1.00 sec? Given:  = 3.14  10 4 rads/sec;  t = 1 sec;  = 0 Want . Dental Drill Example

35. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 35 A high speed dental drill is rotating at 3.14  10 4 rads/sec. What is that in rpm’s? Dental Drill Example

36. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 36 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? v X Car Example

37. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 37 Artificial Gravity A large rotating cylinder in deep space (g  0).

38. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 38 x y N x y N Apply Newton’s 2 nd Law to each: Artificial Gravity Bottom position Top position N is the only force acting. It causes the centripetal acceleration.

39. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 39 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). Space Station Example

40. MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/2010 40 Summary A net force MUST act on an object that has circular motion. Radial Acceleration a r =v 2 /r Definition of Angular Quantities ( , , and  ) The Angular Kinematic Equations The Relationships Between Linear and Angular Quantities Uniform and Nonuniform Circular Motion