Chapter 5 Circular Motion

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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 

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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ (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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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!

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

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

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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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.

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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

MFMcGraw-PHY 1401Ch5b-Circular Motion-Revised 6/21/ 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