Centripetal ForceLaw of Action in CirclesConical PendulumRadial Net ForceAcceleration to VelocityPeriod of RevolutionVertical CurveStaying on TrackForce at the TopHorizontal CurveCurves and FrictionSkiddingBankingCentripetal ForceCentripetal ForceLaw of Action in CirclesLaw of Action in CirclesMotion in a circle has a centripetal acceleration.Motion in a circle has a centripetal acceleration.For every acceleration there is a net force.For every acceleration there is a net force.There must be a There must be a centripetal forcecentripetal force..•Points to the center of the circlePoints to the center of the circle•Magnitude is Magnitude is mama = = mvmv22//rrThe The centrifugalcentrifugal force that we describe is just inertia. force that we describe is just inertia.•It points in the opposite direction – to the outsideIt points in the opposite direction – to the outside•It isn’t a real force It isn’t a real forceConical PendulumConical PendulumA 200. g mass hung is from a 50. cm string as a A 200. g mass hung is from a 50. cm string as a conical pendulum. The period of the pendulum in a conical pendulum. The period of the pendulum in a perfect circle is 1.4 s. What is the angle of the perfect circle is 1.4 s. What is the angle of the pendulum? What is the tension on the string?pendulum? What is the tension on the string?FTRadial Net ForceRadial Net ForceThe mass has a downward The mass has a downward gravitational force, -gravitational force, -mgmg..There is tension in the string.There is tension in the string.•The vertical component must The vertical component must cancel gravitycancel gravityFFTyTy = = mgmgFFTT = = mgmg / cos / cos Tension:Tension: F FTT = = mgmg / cos / cos  = 2.0 N = 2.0 NCentripetal force:Centripetal force:FFTrTr = = mgmg sin sin  / cos / cos = = mgmg tan tan  mgFTFT cos FT sin Acceleration to VelocityAcceleration to VelocityThe acceleration and The acceleration and velocity on a circular path velocity on a circular path are related.are related.mgFTmg tan tantan/tan/22grvgrvrvagmFarPeriod of RevolutionPeriod of RevolutionThe pendulum period is The pendulum period is related to the speed and related to the speed and radius.radius.FTmg tan rLgTgLTLgrvTLTrvLr22222224/coscos/sinsin/sin4tan/sin2/2sinLcos = 0.973 = 13°Vertical CurveVertical CurveA loop-the-loop is a popular A loop-the-loop is a popular rollercoaster feature.rollercoaster feature.There are only two forces There are only two forces acting on the moving car.acting on the moving car.•GravityGravity•Normal forceNormal forceThere is a centripetal There is a centripetal acceleration due to the loop.acceleration due to the loop.•Not uniform circular motionNot uniform circular motionFgFNStaying on TrackStaying on TrackIf the normal force becomes If the normal force becomes zero, the coaster will leave zero, the coaster will leave the track in a parabolic the track in a parabolic trajectory.trajectory.•Projectile motionProjectile motionAt any point there must be At any point there must be enough velocity to maintain enough velocity to maintain pressure of the car on the pressure of the car on the track.track.FgForce at the TopForce at the TopThe forces of gravity and the The forces of gravity and the normal force are both normal force are both directed down.directed down.Together these must match Together these must match the centripetal force.the centripetal force.The minimum occurs with The minimum occurs with almost no normal force.almost no normal force.The maximum is at the The maximum is at the bottom: bottom: aa = = vv22 / / r.r.FgFNgrvgrmrFvmgFrmvFFFNNgNcmin2/Horizontal CurveHorizontal CurveA vehicle on a horizontal curve has a centripetal A vehicle on a horizontal curve has a centripetal acceleration associated with the changing direction.acceleration associated with the changing direction.The curve doesn’t have to be a complete circle.The curve doesn’t have to be a complete circle.•There is still a radius (There is still a radius (rr) associated with the curve) associated with the curve•The force is still The force is still FFcc = = mvmv22//r r directed inwarddirected inwardrFcCurves and FrictionCurves and FrictionOn a turn the force of static friction provides the On a turn the force of static friction provides the centripetal acceleration.centripetal acceleration.In the force diagram there is no other force acting in In the force diagram there is no other force acting in the centripetal direction.the centripetal direction.rFcmgFFFrmvFsffcc /2SkiddingSkiddingThe limit of steering in a The limit of steering in a curve occurs when the curve occurs when the centripetal acceleration centripetal acceleration equals the maximum static equals the maximum static friction. friction. A curve on a dry road (A curve on a dry road (ss = = 1.0) is safe at a speed of 90 1.0) is safe at a speed of 90 km/h.km/h.What is the safe speed on What is the safe speed on the same curve with ice (the same curve with ice (ss = = 0.2)?0.2)?•90 km/h = 25 m/s90 km/h = 25 m/s•rrdrydry = = vv22/ / ss gg = 64 m = 64 m•vv22icyicy = = ss gg r r = 120 m= 120 m22/s/s22•vvicyicy = 11 m/s = 40 km/h = 11 m/s = 40 km/hgvrgrvgrvmgFrmvFssssfc///2maxmax2max2maxBankingBankingCurves intended for Curves intended for higher speeds are higher speeds are banked.banked.Without friction a curve Without friction a curve banked at an angle banked at an angle  can can supply a centripetal force supply a centripetal force FFc c = = mg mg tan tan ..The car can turn without The car can turn without any friction.any