PHYS 1443 – Section 003 Lecture #9Forces of FrictionExample w/ FrictionNewton’s Second Law & Uniform Circular MotionExample of Uniform Circular MotionExample of Banked HighwayForces in Non-uniform Circular MotionExample of Non-Uniform Circular MotionMotion in Resistive ForcesResistive Force Proportional to SpeedNewton’s Law of Universal GravitationMore on Law of Universal GravitationFree Fall Acceleration & Gravitational ForceExample for Gravitational ForceWednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu1PHYS 1443 – Section 003Lecture #9•Forces of Friction•Uniform and Non-uniform Circular Motions•Resistive Forces and Terminal Velocity•Newton’s Law of Gravitation•Kepler’s LawWednesday, Sept. 24, 2003Dr. Jaehoon YuRemember the first term exam on next Monday, Sept. 29!!Homework #5 due at midnight next Thursday, Oct. 2!!Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu2Forces of FrictionnfssResistive force exerted on a moving object due to viscosity or other types frictional property of the medium in or surface on which the object moves.Force of static friction, fs:Force of kinetic friction, fkThe resistive force exerted on the object until just before the beginning of its movementThe resistive force exerted on the object during its movementnfkkThese forces are either proportional to velocity or normal forceEmpirical Formula What does this formula tell you? Frictional force increases till it reaches to the limit!!Beyond the limit, there is no more static frictional force but kinetic frictional force takes it over.Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu3Example w/ FrictionSuppose a block is placed on a rough surface inclined relative to the horizontal. The inclination angle is increased till the block starts to move. Show that by measuring this critical angle, c, one can determine coefficient of static friction, s.FFree-bodyDiagramxyMaFgnnF= -Mgfs=knyFxFsNet forcex comp.y comp.sfnxyaMsgfnF sgxfFsfMgsin0nscMgsinyMagyFncMgncos0gyFcMgcosnMgcsinccMgMgcossinctanWednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu4Newton’s Second Law & Uniform Circular MotionFrFrrmrvar2The centripetal acceleration is always perpendicular to velocity vector, v, for uniform circular motion.The force that causes the centripetal acceleration acts toward the center of the circular path and causes a change in the direction of the velocity vector. This force is called centripetal force.Are there forces in this motion? If so, what do they do?rFWhat do you think will happen to the ball if the string that holds the ball breaks? Why?Based on Newton’s 1st law, since the external force no longer exist, the ball will continue its motion without change and will fly away following the tangential direction to the circle.rmarvm2Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu5Example of Uniform Circular MotionA ball of mass 0.500kg is attached to the end of a 1.50m long cord. The ball is moving in a horizontal circle. If the string can withstand maximum tension of 50.0 N, what is the maximum speed the ball can attain before the cord breaks? mFrrvar2Centripetal acceleration:rFWhen does the string break?When the centripetal force is greater than the sustainable tension.Trvm 2Calculate the tension of the cord when speed of the ball is 5.00m/s. NrvmT 33.85.100.5500.022 smmTrv /2.12500.05.10.50rmarvm2TWednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu6Example of Banked Highway(a) For a car traveling with speed v around a curve of radius r, determine a formula for the angle at which a road should be banked so that no friction is required to keep the car from skidding. rmvn2sin yFxFsmhrkmv /14/50 x comp.y comp.xyrmansinsinmgn 0 mgncosrmvmgmgn2tancossinsin grv2tan 4.08.95014tan2rmvn2sin0mgn cos(b) What is this angle for an expressway off-ramp curve of radius 50m at a design speed of 50km/h? o224.0tan1Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu7Forces in Non-uniform Circular MotionThe object has both tangential and radial accelerations.What does this statement mean?The object is moving under both tangential and radial forces.FrFtFtrFFF These forces cause not only the velocity but also the speed of the ball to change. The object undergoes a curved motion under the absence of constraints, such as a string. 22traaa How does the acceleration look?Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu8Example of Non-Uniform Circular MotionA ball of mass m is attached to the end of a cord of length R. The ball is moving in a vertical circle. Determine the tension of the cord at any instant when the speed of the ball is v and the cord makes an angle with vertical. TmWhat are the forces involved in this motion?tFThe gravitational force Fg and the radial force, T, providing tension. RFg=mgAt what angles the tension becomes maximum and minimum. What are the tensions?singatrFcos2gRvmTtangential comp.Radial comp.sinmgtmacosmgTrmaRvm2Wednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu9Motion in Resistive ForcesMedium can exert resistive forces on an object moving through it due to viscosity or other types frictional property of the medium.These forces are exerted on moving objects in opposite direction of the movement. Some examples? These forces are proportional to such factors as speed. They almost always increase with increasing speed. Two different cases of proportionality: 1. Forces linearly proportional to speed: Slowly moving or very small objects2. Forces proportional to square of speed: Large objects w/ reasonable speedAir resistance, viscous force of liquid, etcWednesday, Sept. 24, 2003 PHYS 1443-003, Fall 2003Dr. Jaehoon Yu10Resistive Force Proportional to Speed Since the resistive force is proportional to speed, we can write R=bvmLet’s consider that a ball of mass m is falling through a liquid.RmgvRFFgThis equation also tells you that0 when , vgvmbgdtdvThe above equation also tells us that as time goes on
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