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Circular Motion The Radian Objects moving in circular (or nearly circular) paths are often measured in radians rather than degrees. In the diagram, the angle θ, in radians, is defined as follows So, if s = r the angle is 1 rad and if s is equal to the full circumference of the circle, the angle is 2π rad. (In other words, 360° = 2π rad.) Radians and Degrees Angular Displacement (θ) If we consider a small body to be moving round the circle from A to B we say that it has experienced an angular displacement of θ radians. The relation between the (linear) distance moved, d, of the body and the angular displacement θ, is given by d = rθ Also, if the angle is small, d is very nearly equal to the magnitude of the linear displacement of the body. Angular Velocity (ω) Suppose that the body moved from P1 to P2 in a time t. The linear speed, v, of the body is given by v = d/t. If we divide d=rθ by t, we have The angular velocity ω is defined as Units of ω are rad/s Therefore, v = rω Merry-Go Round http://animations.50webs.com/free_cartoons_mobile_animation.htm What happens to your speed as you go to the middle of a merry-go round? a. The speed remains constant. b. The speed increases c. The speed decreases Merry-Go Round http://animations.50webs.com/free_cartoons_mobile_animation.htm What happens to your angular speed as you go to the middle of a merrygo round? a. The angular speed remains constant. b. The angular speed increases c. The angular speed decreases http://mocoloco.com/art/upload/2009/12/biondo_merry_go_round.jpg Angular Acceleration(α) Previously we assumed that the body moved from P1 to P2 with constant speed. If the linear speed of the body changes then, obviously, the angular speed (velocity) also changes. The angular acceleration, α, is the rate of change of angular velocity. So, if the angular velocity changes uniformly from ω1 to ω2 in time t, then we can write: Now, linear acceleration, a, is given by Substituting v=rω We find, a r a v f vo t a v f vo t Circular Motion Definitions Time Period, T The time period of a circular motion is the time taken for one revolution. Rotational Frequency, f The rotational frequency of a circular motion is the number of revolutions per unit time. Time period is the inverse of frequency Also, and Rotational Motion What is the relationship between Liner and Rotational motion quantities? Acceleration Equations Revisited What is the Acceleration? a. b. c. d. No acceleration Acceleration outward Acceleration toward the center of the circle Acceleration points tangential to the circle Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. A plane attached to a string flies in a circle at constant speed Even though the speed is constant, velocity is not constant since the direction is changing: acceleration! If released the plane would travel at a constant speed tangentially. Therefore, the change in velocity would be a vector from A to P If θ is very small, this acceleration (Δv/t) points to the center of the circle. This is called centripetal acceleration Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. What is the Acceleration? Centripetal Acceleration Even though the speed is constant, velocity is not constant since the direction is changing: must be some acceleration! Consider average acceleration in time Δt a avg v t As we shrink Δ t, Δv / Δt v v v2 R dv / dt = a v1 v2 v1 R v v t seems like v v (hence v v/t ) points at the origin! Centripetal Acceleration v v2 R We see that a points in the - R direction. But R = vt for small t v1 v vt So: v R a = dv v / dt v R Similar triangles: v R v2 R R v v 2 t R v2 Magnitude: a R Direction:- r (toward center of circle) 2r Since and v = ωr v2 a = ω a R Centripetal Acceleration http://www.lyon.edu/webdata/users/shutton/phy240-fall2003/circle1.gif Centripetal Acceleration (ac) ‘Centripetal’ means center seeking Vector direction always points toward the center of the circle Magnitude: 2 v ac r v = speed r = radius of the circle http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/ucm.gif Centripetal Acceleration (ac) http://physics.csustan.edu/Astro/Help/NEWTON/cpetal2.gif What is the Net Force? a. b. c. d. No net force Net Force points outward Net Force points toward the center of the circle Net Force points tangential to the circle Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. A plane attached to a string flies in a circle at constant speed a. b. c. d. No net force Net Force points outward Net Force points toward the center of the circle Net Force points tangential to the circle Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. What is the Net Force? A plane attached to a string flies in a circle at constant speed Centripetal Force, Fc, is this net force. Recall, Newton's 2nd Law Fnet=ma Therefore, Fc = mac 2 v Fc m r Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. What is the Net Force? A plane attached to a string flies in a circle at constant speed acts toward the center of the circle depends on mass, speed, and size of the circle 2 v Fc m r Net force, provided by another force or interactions of forces http://motivate.maths.org/conferences/conf14/images/circular_motion3.gif Centripetal Force (Fc) Date Physics http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/rht.html Centripetal Acceleration Example When you are driving a car, and you turn the steering wheel sharply to the right in order to turn the car to the right, you "feel" as if a force is pushing your body to the left against the door. In order for your body to follow the car in the tight circular path, something has to push your body toward the center of the circle-- in this case it is the driver's-side door-- and your tendency otherwise is to travel in a straight line tangent to the circular path Centripetal Force Source What provides the centripetal force (Fc) in the following scenarios? A car going around a corner at a constant speed. Friction force between the tires and the pavement Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. Centripetal Force Source What provides the centripetal force (Fc) in the following scenarios? A ball on a string being twirled in a circle. Tension force of the string on the ball http://www.mansfieldct.org/schools/mms/staff/ha nd/lawsCentripetalForce_files/image004.jpg http://www.vast.org/vip/book/LOOPS/HOME7.GIF Centripetal Force Source The sun’s force of gravity on the planets. http://quest.nasa.gov/aero/planetary/orbit/Image1.jpg What provides the centripetal force (Fc) in the following scenarios? The planets orbiting around the sun. Centripetal Force Source What provides the centripetal force (Fc) in the following scenarios? A motorcycle stuntman going around a loop. Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. Both the normal force of the track and gravity. Centripetal Force Example A ball attached to a string is twirled in a circle of radius 0.5 m with a speed of 2 m/s. If the ball has a mass of 0.25 kg, what is the tension of the string. Knowns: http://www.frontiernet.net/~jlkeefer/centacc.gif m = 0.25 kg v = 2 m/s r = 0.5 m F T F n et F c 2 v Fc m r 0 .252 T Fc 0 .5 2 T=2N Moving in a Straight line on a Horizontal Surface Moving in a Straight line on a Horizontal Surface The normal reaction, FN, has no component acting towards the center of the circular path. Therefore the required centripetal acceleration is provided by the force of friction, Ff, between the wheel and the road. If the force of friction is not strong enough, the vehicle will skid. Turning on a Banked Surface The normal reaction, FN, now has a component acting towards the center of the circular path. Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. If the angle, q, is just right, the correct centripetal acceleration can be provided by the horizontal component of the normal reaction. This means that, even if there is very little force of friction the vehicle can still go round the curve with no tendency to skid. Angle of Banking The magnitude of the horizontal component of the normal force is FN x FN sin q This force causes the centripetal acceleration, so, the magnitude of NX is also given by mv2 FN x r So, F sin q mv 2 N r The vertical forces acting on the vehicle are in equilibrium. Therefore, summing the vertical forces Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. FN cos q mg Angle of Banking Solving into FN cos q mg mv FN sin q r Simplifying, 2 for FN and substituting mg sin q mv 2 gives: cos q r v2 tan q rg Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. This equation allows us to calculate the angle θ needed for a vehicle to go round the curve at a given speed, v, without any tendency to skid. NASCAR Physics What is the minimum speed a NASCAR car must go not to slide down the banked curves of the Bristol Motor Speedway if the banked angle is 36° and the radius is 73.5m? Loopy The minimum speed to complete a loop requires: Speed large enough to reach the top of the loop. At the top of the loop Fnet = Fg + FN For the minimum speed FN = 0 2 mv Therefore, F F F min recall Fg mg g net c r So, vmin gr at the top of the loop Cutnell & Johnson, Wiley Publishing, Physics 5th Ed. Kepler’s Law of Periods Proof Assumptions: • Must conform to equations for circular motion • Newton’s Universal Law of Gravity • Planet rotates in a circular (elliptical) path Fnet mv 2 Fc mac r Newton’s 3rd Law symmetry Fgravity Fnet Fc Recall, GMm mv 2 Fg 2 r r so 2 T v r Therefore, GMm m 4 r 2 r2 T2 2r T Law of Periods T 4 rearranging 3 R GM 2 2 Testing the Inverse Square Law of Gravitation The acceleration due to gravity at the surface of the earth is 9.8m/s2. If the inverse square relationship for gravity (Fg~1/r2) is correct , then, at a distance ~60 times further away from the center of the earth, the 9.8 m 2.72 10 acceleration due to gravity should be 60 s The centripetal acceleration of the moon is given by a vr where the radius of the moon’s orbit is r = 3.84 × 108 m and the time period of the moon’s 2r orbital motion is T = 27.3 days. v 3 2 2 c ac 4 r T2 2 ac 4 2 (3.84 10 8 ) 24h (3600s) (27.3days)(1day ) 1h T 2 a c 2.72 10 3 m s2 2