PHYS 1443 Section 001 Lecture 6 Wednesday June 7 2006 Dr Jaehoon Yu Application of Newton s Laws Motion without friction Forces of Friction Uniform and Non uniform Circular Motions Resistive Forces and Terminal Velocity Newton s Law of Universal Gravitation Kepler s Laws Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 1 Announcements Quiz tomorrow Thursday June 8 At the beginning of the class CH 1 what we cover today CH 6 4 or 5 Reading assignments CH 4 7 5 4 5 5 Mid term exam 8 00 10am Thursday June 15 in class CH 1 8 or 9 Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 2 Example w o Friction A crate of mass M is placed on a frictionless inclined plane of angle a Determine the acceleration of the crate after it is released y y M n n Fx Ma x Fgx Mg sin Free body Diagram x Fg xa xd y r r r r F Fg n ma Supposed the crate was released at the top of the incline and the length of the incline is d How long does it take for the crate to reach the bottom and what is its speed at the bottom a x g sin F Mg Fy Ma y n Fgy n mg cos 0 2d 1 2 1 2 d vixt a x t g sin t t 2 2 g sin vxf vix a xt g sin 2d 2dg sin g sin v xf 2dg sin Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 3 Forces of Friction Resistive 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 These forces are either proportional to the velocity or the normal force Force of static friction fs The resistive force exerted on the object until just before the r r Empirical beginning ofFrictional its movement f m n What does force increases till it Formula s s this formula reaches the limit tell you Beyond the limit the object moves and there is NO MORE static friction but kinetic friction takes it over Force of kinetic friction fk r r f k mk n The resistive force exerted on the object during its movement Which direction does kinetic friction apply Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu Opposite to the motion 4 Example w Friction Suppose 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 y y n M a n Free body fs kn Diagram x x Fg F Mg r r r r r Net force F Ma Fg n f s x comp Fx Fgx f s Mg sin f s 0 f s s n Mg sin c y comp Fy Ma y n Fgy n Mg cos c 0 n Fgy Mg cos c Mg sin c Mg sin c tan c s Mg cos c n Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 5 Newton s Second Law Uniform Circular Motion The centripetal acceleration is always perpendicular to velocity 2 vector v for uniform v circular ar motion r Are there forces in this motion If so what doacceleration they do The force that causes the centripetal acts toward the center of the circular path and causes the change in the direction of the velocity vector This force is called the centripetal force v2 Fr mar m r What do you think will happen to the ball if the string that holds the ball breaks Based on Newton s 1st law since the external force no longer exist Therefore the ball will continue its motion without change and will fly away following the tangential direction to the circle Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 6 Example of Uniform Circular Motion A 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 Fr v2 ar r Centripet m al accelerat When v2 ion does Fr mar m T the string r break when the centripetal force is greater than the sustainable tension 2 v Tr 50 0 1 5 12 2 m s m T v m 0 500 r Calculate the tension of the cord when speed of the ball is 5 00m s Wednesday June 7 2006 v2 5 00 8 33 N 0 500 T m r 1 5 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu 2 7 Example 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 y x 2 mv x comp Fx FN sin q mar FN sin q 0 2 r mv FN sin q r FN cos q mg y comp Fy FN cos q mg 0 2 mg mv mg sin q FN sin q FN mg tan q sin q cos q r v2 tan gr b What is this angle for an expressway off ramp curve of radius 50m at a design speed of 50km h v 50km hr 14m s Wednesday June 7 2006 tan 14 2 50 9 8 0 4 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu tan 1 0 4 22o 8 Forces in Non uniform Circular Motion The object has both tangential and radial accelerations What does this statement mean Fr F Ft The object is moving under both tangential and radial forces r r r F Fr Ft 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 What is the magnitude of the net acceleration Wednesday June 7 2006 PHYS 1443 001 Summer 2006 Dr Jaehoon Yu a ar2 at2 9 Example for Non Uniform Circular Motion A 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 T R m What are the forces involved in this motion The gravitational force Fg The radial force T providing tension Fg mg tangential comp Radial comp F t mg sin mat v2 Fr T mg cos mar m R at g sin v2 T m g cos R At what angles the tension becomes maximum and minimum What are the tensions Wednesday June 7 2006 PHYS …
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