MFMcGrawCh5-Circular Motion-Revised 2/15/103 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 ii ff
MFMcGrawCh5-Circular Motion-Revised 2/15/104 x y ii ff r arc length = s = r is a ratio of two lengths; it is a dimensionless ratio! Arc Length
MFMcGrawCh5-Circular Motion-Revised 2/15/105 The average and instantaneous angular velocities are: is measured in rads/sec. Angular Speed
MFMcGrawCh5-Circular Motion-Revised 2/15/106 An object moves along a circular path of radius r; what is its average speed? Also, (instantaneous values). x y ii ff r Angular Speed
MFMcGrawCh5-Circular Motion-Revised 2/15/107 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
MFMcGrawCh5-Circular Motion-Revised 2/15/108 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
MFMcGrawCh5-Circular Motion-Revised 2/15/109 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1010 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1011 The magnitude of the radial acceleration is: Centripetal Acceleration
MFMcGrawCh5-Circular Motion-Revised 2/15/1012 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1013 (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
MFMcGrawCh5-Circular Motion-Revised 2/15/1014 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1015 From (2) Solving for r: What is ? Unbanked Curve
MFMcGrawCh5-Circular Motion-Revised 2/15/1016 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.
MFMcGrawCh5-Circular Motion-Revised 2/15/1017 FBD for the car: x y N w Apply Newton’s Second Law: Banked Curves
MFMcGrawCh5-Circular Motion-Revised 2/15/1018 Rewrite (1) and (2): Divide (1) by (2): Banked Curves
MFMcGrawCh5-Circular Motion-Revised 2/15/1019 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: Solve for the speed of the satellite:
MFMcGrawCh5-Circular Motion-Revised 2/15/1020 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1021 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. Circular Orbits
MFMcGrawCh5-Circular Motion-Revised 2/15/1022 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!
MFMcGrawCh5-Circular Motion-Revised 2/15/1023 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 perpendicualr and parallel components.
MFMcGrawCh5-Circular Motion-Revised 2/15/1024 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1025 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1026 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1027 Linear and Angular Acceleration The average and instantaneous angular acceleration are: is measured in rads/sec 2.
MFMcGrawCh5-Circular Motion-Revised 2/15/1028 Recalling that the tangential velocity is v t = r means the tangential acceleration is atat Linear and Angular Acceleration
MFMcGrawCh5-Circular Motion-Revised 2/15/1029 Linear (Tangential) Angular With Linear and Angular Kinematics “a” and “a t ” are the same thing
MFMcGrawCh5-Circular Motion-Revised 2/15/1030 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1031 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
MFMcGrawCh5-Circular Motion-Revised 2/15/1032 Artificial Gravity A large rotating cylinder in deep space (g 0).
MFMcGrawCh5-Circular Motion-Revised 2/15/1033 x y N x y N FBD for the person Apply Newton’s 2 nd Law to each: Artificial Gravity Bottom position Top position
MFMcGrawCh5-Circular Motion-Revised 2/15/1034 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 ) = Hz (or 2.7 rpm). Space Station Example
MFMcGrawCh5-Circular Motion-Revised 2/15/1035 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