Lecture 10Uniform Circular MotionNon-uniform Circular MotionCircular motionKey stepsExample The pendulumExample Gravity, Normal Forces etc.Conical Pendulum (very different)Loop-the-loop 1Loop-the-loop 1Loop-the-loop 2Loop-the-loop 3Example, Circular Motion Forces with Friction (recall mar = m |vT | 2 / r Ff ≤ ms N )ExampleAnother ExampleUCM: Acceleration, Force, VelocityZero Gravity RideBanked CurvesSlide 20Banked Curves, Testing your understandingLocomotion: how fast can a biped walk?How fast can a biped walk?Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Orbiting satellites vT = (gr)½Geostationary orbitSlide 32Physics 207: Lecture 10, Pg 1Lecture 10Goals:Goals: Employ Newton’s Laws in 2D problems with circular motionAssignment: HW5, (Chapters 8 & 9, due 3/4, Wednesday)For Tuesday: Finish reading Chapter 8, start Chapter 9.Physics 207: Lecture 10, Pg 2Uniform Circular MotionFor an object moving along a curved trajectory, with non-uniform speeda = ar (radial only)arv|ar |= vT2rPerspective is importantPhysics 207: Lecture 10, Pg 3Non-uniform Circular MotionFor an object moving along a curved trajectory, with non-uniform speeda = ar + at (radial and tangential)aratdtd| |v|aT |= |ar |= vT2rPhysics 207: Lecture 10, Pg 4Circular motionCircular motion implies one thing|aradial | =vT2 / rPhysics 207: Lecture 10, Pg 5Key stepsIdentify forces (i.e., a FBD)Identify axis of rotationApply conditions (position, velocity, acceleration)Physics 207: Lecture 10, Pg 6Example The pendulumConsider a person on a swing: When is the tension on the rope largest? And at that point is it :(A) greater than(B) the same as(C) less thanthe force due to gravity acting on the person?axis of rotationPhysics 207: Lecture 10, Pg 7Example Gravity, Normal Forces etc.vTmgTat bottom of swing vT is maxFr = m ac = m vT2 / r = T - mgT = mg + m vT2 / rT > mgmgTat top of swing vT = 0Fr = m 02 / r = 0 = T – mg cos T = mg cos T < mgaxis of rotationyxPhysics 207: Lecture 10, Pg 8Conical Pendulum (very different)Swinging a ball on a string of length L around your head(r = L sin )axis of rotation Fr = mar = T sin Fz = 0 = T cos – mg so T = mg / cos (> mg)mar = mg sin / cos ar = g tan vT2/r vT = (gr tan )½ Period:t = 2 r / vT =2 (r cot /g)½Physics 207: Lecture 10, Pg 9A match box car is going to do a loop-the-loop of radius r. What must be its minimum speed vt at the top so that it can manage the loop successfully ?Loop-the-loop 1Physics 207: Lecture 10, Pg 10To navigate the top of the circle its tangential velocity vT must be such that its centripetal acceleration at least equals the force due to gravity. At this point N, the normal force, goes to zero (just touching). Loop-the-loop 1Fr = mar = mg = mvT2/r vT = (gr)1/2mgvTPhysics 207: Lecture 10, Pg 11The match box car is going to do a loop-the-loop. If the speed at the bottom is vB, what is the normal force, N, at that point?Hint: The car is constrained to the track. Loop-the-loop 2mgv NFr = mar = mvB2/r = N - mgN = mvB2/r + mgPhysics 207: Lecture 10, Pg 12Once again the car is going to execute a loop-the-loop. What must be its minimum speed at the bottom so that it can make the loop successfully?This is a difficult problem to solve using just forces. We will skip it now and revisit it using energy considerations later on…Loop-the-loop 3Physics 207: Lecture 10, Pg 14Example, Circular Motion Forces with Friction (recall mar = m |vT | 2 / r Ff ≤ s N )How fast can the race car go? (How fast can it round a corner with this radius of curvature?)mcar= 1600 kgS = 0.5 for tire/road r = 80 m g = 10 m/s2rPhysics 207: Lecture 10, Pg 15Only one force is in the horizontal direction: static friction x-dir: Fr = mar = -m |vT | 2 / r = Fs = -s N (at maximum)y-dir: ma = 0 = N – mg N = mgvT = (s m g r / m )1/2vT = (s g r )1/2 = ( x 10 x 80)1/2vT = 20 m/s ExampleNmgFsmcar= 1600 kgS = 0.5 for tire/road r = 80 m g = 10 m/s2yxPhysics 207: Lecture 10, Pg 16A horizontal disk is initially at rest and very slowly undergoes constant angular acceleration. A 2 kg puck is located a point 0.5 m away from the axis. At what angular velocity does it slip (assuming aT << ar at that time) if s=0.8 ?Only one force is in the horizontal direction: static friction x-dir: Fr = mar = -m |vT | 2 / r = Fs = -s N (at )y-dir: ma = 0 = N – mg N = mgvT = (s m g r / m )1/2vT = (s g r )1/2 = ( x 10 x 0.5)1/2vT = 2 m/s = vT / r = 4 rad/sAnother Examplempuck= 2 kgS = 0.8 r = 0.5 m g = 10 m/s2yxNmgFsPhysics 207: Lecture 10, Pg 17UCM: Acceleration, Force, VelocityFavPhysics 207: Lecture 10, Pg 18Zero Gravity RideA rider in a “0 gravity ride” finds herself stuck with her back to the wall.Which diagram correctly shows the forces acting on her?Physics 207: Lecture 10, Pg 19Banked CurvesIn the previous car scenario, we drew the following free body diagram for a race car going around a curve on a flat track.nmgFfWhat differs on a banked curve?Physics 207: Lecture 10, Pg 20Banked CurvesFree Body Diagram for a banked curve.Use rotated x-y coordinatesResolve into components parallel and perpendicular to bankNmgFfFor very small banking angles, one can approximate that Ff is parallel to mar. This is equivalent to the small angle approximation sin = tan , but very effective at pushing the car toward the center of the curve!!marxx y y Physics 207: Lecture 10, Pg 21Banked Curves, Testing your understandingFree Body Diagram for a banked curve.Use rotated x-y coordinatesResolve into components parallel and perpendicular to bankNmgFfAt this moment you press the accelerator and, because of the frictional force (forward) by the tires on the road you accelerate in that direction.How does the radial acceleration change? maryy x x Physics 207: Lecture 10, Pg 22Locomotion: how fast can a biped walk?Physics 207: Lecture 10, Pg 23How fast can a biped walk?What about weight?(a) A heavier person of equal height and proportions can walk faster than a lighter person(b) A lighter person of equal height and proportions can walk faster than a heavier person(c) To first order, size doesn’t matterPhysics 207: Lecture 10, Pg 24How fast can a biped walk?What about height?(a) A taller person of equal weight and proportions can walk faster than a shorter person(b) A shorter person
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