PS113 Chapte 5 Dynamics of Unifom Cicula Motion 1 Unifom cicula motion Unifom cicula motion is the motion of an object taveling at a constant (unifom) speed on a cicula path. The peiod T is the time equied to tavel once aound the cicle (i.e., one complete evolution). v = 2π T 2 Centipetal acceleation Using simila tiangles, one can show that: v v = v t a c = v t = v2 whee the centipetal acceleation vecto always points towad the cente of the cicle and continually changes diection as the object moves. 1
Poblem 9: Compute-contolled display sceens povide dives in the Indianapolis 500 with a vaiety of infomation about how thei cas ae pefoming. Fo instance, as a ca is going though a tun, a speed of 221 mi/h (98.8 m/s) and a centipetal acceleation of 3.00g (thee times the acceleation due to gavity) ae displayed. Detemine the adius of the tun (in metes). 332 m 3 Centipetal foce The centipetal foce is the name given to the net foce equied to keep an object of mass m, moving at a speed v, on a cicula path of adius and has a magnitude of: F c = mv2 Diection: The centipetal foce always points towad the cente of the cicle and continually changes diection as the object moves. The centipetal foce does not denote a new foce to use on the left-hand side of the F = ma equation. On the contay, F c is the esultant foce that goes on the ight-hand side Poblem 14: At an amusement pak thee is a ide in which cylindically shaped chambes spin aound a cental axis. People sit in seats facing the axis, thei backs against the oute wall. At one instant the oute wall moves at a speed of 3.2 m/s, and an 83-kg peson feels a 560-N foce pessing against his back. What is the adius of a chambe? Answe: 1.52 m 2
Poblem 18: A ca is safely negotiating an unbanked cicula tun at a speed of 21 m/s. The oad is dy, and the maximum static fictional foce acts on the ties. Suddenly a long wet patch in the oad deceases the maximum static fictional foce to one-thid of its dy-oad value. If the ca is to continue safely aound the cuve, to what speed must the dive slow the ca? 4 Banked cuves When a ca tavels without skidding aound an unbanked cuve, the static fictional foce between the ties and the oad povides the centipetal foce. At a paticula speed, the need fo fiction can be eliminated completely if the cuve is banked at an angle elative to the hoizontal. Using a fee-body diagam to descibe the foces acting on a ca taveling on a banked cuve, we find: F c = F N sin θ = m v2 Since the vetical component of F N must balance the weight of the ca, we have that F N cos θ = mg. Using these two equations we find that: F N sin θ F N cos θ = m v2 / mg tan θ = v2 g 3
Fo a given and v, this equations descibes the angle at which the tack must be banked so no static fiction is equied. Poblem 24: On a banked ace tack, the smallest cicula path on which cas can move has a adius of 112 m, while the lagest has a adius of 165 m, as the dawing illustates. The height of the oute wall is 18 m. Find (1) the smalls and (b) the lagest speed at which cas can move on this tack without elying on fiction. 4
5 Satellites in cicula obits Thee is only one speed that a satellite can have if it is to emain in a cicula obit (i.e., with a fixed adius). Using Newton s 2 nd law of motion, we find: F c = G m M E 2 = mv2 whee is the distance fom the cente of the eath to the satellite. Solving fo the speed of the satellite, v, we find: v = G M E Fo a given obit, a satellite with a lage mass has exactly the same obital speed as a satellite with a small mass. Geosynchonous satellites taveling about the eath once evey 24 hous ae all at the same altitude. 5
6 Appaent weightlessness and atificial gavity Astonauts in the Intenational Space Station o the space shuttle expeience a state of appaent weightlessness because it is simila to the condition of zeo appaent weight occuing in an elevato duing fee-fall. One can geneate atifical gavity by otating the local envionment while in space. Cetain designs fo space stations and ocket ships include this featue of geneating atificial gavity. a c = v2 = (2π/T )2 = 4π2 T 2 Poblem 32: A ocket is used to place a synchonous satellite in obit about the eath. What is the speed of the satellite in obit? Answe: 3070 m/s 6
7 Vetical cicula motion The concept of vetical cicula motion only makes sense in a gavitational field (i.e., the vetical diection needs a efeence, namely the diection along the line of gavity). Again, we e only going to look at unifom cicula motion whee the centipetal foce (the esultant foce) is F C = mv 2 /. The question that s usually posed is the following: What is the nomal foce N (o tension, T ) equied to keep an object in unifom cicula motion? The answe to this will depend upon the location of the paticle while in motion. We will need to use Newton s 2 nd law to calculate the magnitude of the nomal foce. At the top of the motion, we have: F = mac F N mg = m v2 Solving this equation fo F N, the nomal foce, we have F N = mv 2 / mg. The intepetation of this equation is the following: In ode fo the nomal foce to exist at the top of the motion, the quantity mv 2 / must be at least as lage as mg. Negative values of the nomal foce means that that thee is no nomal foce (o tension). At the bottom of the motion, we have: F = mac +F N mg = m v2 Solving this equation fo F N, the nomal foce, we have F N = mv 2 /+mg. The intepetation of this equation is staight-fowad because the nomal foce is always positive at the bottom of the motion. 7
Poblem 44: A special electonic senso is embedded in the seat of a ca that takes ides aound a cicula loop-the-loop ide at an amusement pak. The senso measues the magnitude of the nomal foce that the seat exets on a ide. The loop-the-loop ide is in the vetical plane and its adius is 21 m. Sitting on the seat befoe the ide stats a ide is level and stationay, and the electonic senso eads 770 N. At the top of the loop the ide is upside down and moving, and senso eads 350 N. What is the speed of the ide at the top of the loop? 8