VIEWS: 20 PAGES: 14 CATEGORY: Politics & History POSTED ON: 6/1/2010 Public Domain
Lecture 5: Dynamics of Uniform Circular Motion Uniform Circular Motion DEFINITION OF UNIFORM CIRCULAR MOTION Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. Period, T = Time required to travel once around the circle or complete one revolution In UNIFORM circular motion, the “speed”- magnitude of the velocity vector is a CONSTANT But the direction is NOT A CONSTANT Centripital Acceleration •“Centripital” – Center Seeking • Since the velocity vector under uniform circular motion changes, there must be an acceleration – CENTRIPITAL acceleration (ac) . • Magnitude of ac depends on the speed of the object (v) and the radius (r) of the circular path. • Magnitude: The centripetal acceleration of an object moving with a speed v on a circular path of radius r has a magnitude ac given by : v2/r • Direction: The centripetal acceleration vector always points toward the center of the circle and continually changes direction as the object moves Centripital Force •Newton’s Second Law: F = ma, when ever there is an acceleration, there should be a net force associated with that. • CENTRIPITAL Force: (Fc) points in the same direction as the centripital acceleration – towards the center of the circle. Magnitude: The centripetal force is the name given to the net force required to keep an object of mass m, moving at a speed v, on a circular path of radius r, and it has a magnitude of Direction: The centripetal force always points toward the center of the circle and continually changes direction as the object moves. Centripital Force – Is this a “NEW” Force? •Does NOT denote a New and Separate force created by nature •It is the net force pointing towards the center of the circular path •This net force is the vector sum of all force components that point in the radial direction Banked Curves mv 2 FC FN Sin r FN Cos mg No Need of friction to hold the car in place Satellites in Circular Orbit • Gravitational force( pull) provides the centripital force • There is only one speed that the satellite can have if it is to remain in an orbit of fixed radius. GM E ms mv 2 FC 2 r r GM E v r • Closer the satellite, the greater nee\\ds to be the velocity. • Note: Mass, ms of the satellite does not appear in the equation => for a given orbit a satellite with large mass has exactly the same orbital speed as a satellite with small mass. Global Positioning Satellites - 24 Satellites - Each satellite has an atomic clock (Remember the need for precision timing) - Cars with GPS receive radio signals from these satellites Evidence of Black Hole at the center of galaxy M87 Period of Satellites in Circular Orbit • Period of a satellite = Time required to make one orbital revolution Geo-Synchronous Satellites GM E v r GM E 2 r r T 3 2 r 2 T GM E 3 Kepler ' s Third Law :T r 2 Apparent Weightlessness and Artificial Gravity the apparent weight in the satellite is zero, just as it is in the freely falling elevator. The only difference between the satellite and the elevator is that the satellite moves on a circle, so that its “falling” does not bring it closer to the earth. In contrast to the apparent weight, the true weight is the gravitational force (F = GmME/r2) that the earth exerts on an object and is not zero in a freely falling elevator or aboard an orbiting satellite. Artificial Gravity The surface of the rotating space station pushes on an object with which it is in contact and thereby provides the centripetal force that keeps the object moving on a circular path. The outer ring (radius = r0) of this rotating space laboratory simulates gravity on earth, while the inner ring (radius = r1) simulates gravity on Mars. Vertical Circular Motion