VISUAL PHYSICS ONLINE MODULE 5 ADVANCED MECHANICS NON-UNIFORM CIRCULAR MOTION Equation of a cicle x y Angula displacement [ad] Angula speed d constant t [ad.s -1 ] dt Tangential velocity v v [m.s -1 ] (diection: tangent to cicle) Centipetal acceleation ac ac v / [m.s - ] (diection: towads cente of cicle / pependicula to cicle) A foce must be applied to an object to give it cicula motion. This net foce is called the centipetal foce. FC F FC m (diection: towads cente of cicle / pependicula to cicle) Vecto poduct C A B AB sin nˆ Toque [N.m] F F sin nˆ F sin F F Fsin
Review: Language of Physics VECTOR (CROSS) PRODUCT The vecto poduct o coss poduct of two vectos A and B is defined as C A B AB sin nˆ The magnitude of the vecto C is C C A B sin. The vecto ˆn is a unit vecto which is pependicula to both the vectos A and B. The angle between the two vectos is always less than o equal to 180 o. The sine ove this ange of angles is neve negative, hence the magnitude of the vecto poduct is always positive o o zeo 0 180 0 sin 1.
The diection of the vecto poduct is pependicula to both the vectos A and B. The diection is given by the ight-hand scew ule. The thumb of the ight hand gives the diection of the vecto poduct as the finges of the ight hand otate fom along the diection of the vecto A towads the diection of the vecto B. TORQUE and the VECTOR PRODUCT What is the physics of opening a doo? It is the toque applied to the doo that is impotant and not the foce. A foce can cause an object to move and a toque can cause an object to otate. A toque is often thought of as a foce multiplied by a distance. Howeve, using the idea of the vecto (coss) poduct we can pecisely define what we mean by the concept of toque. F F sinnˆ
The vecto is the toque applied, the vecto is the leve am distance fom the pivot point to the point of application of the foce F. The angle is the angle between the vectos and F. The diection of the toque ˆn is found by applying the igh-hand scew ule: the thumb points in the diection of the toque as you otate the finges of the ight hand fom along the line of the vecto to the vecto F. The toque is pependicula to both the position vecto and the foce F. The magnitude of the toque can be expessed as F sin F whee F is the component of the applied foce F which acts at ight angles to the adius vecto. F Fsin
A non-zeo net foce acting on an object causes it to acceleate in the diection of the foce (Newton s nd Law of motion). Howeve, a net toque acting on an object is necessay to cause an angula acceleation of an object. In unifom cicula motion, the tangential speed of the object is constant. To change the tangential speed,a non-zeo toque must act on it, poducing a tangential component of the acceleation. The acceleation a of the object has two components the centipetal acceleation a C (diected towads the cente) and a tangential component a T (diected along the tangent to the cicle).
Example A ball of mass 0.50 kg at the end of a long sting of length.3 m (negligible mass) is swung in a vetical cicle. Detemine the minimum speed the ball must have at the top of its ac. Calculate the sting tension at the bottom of the swing if the ball is moving at twice the speed of the ball had at the top of the swing. Solution Visualize the poblem / how to appoach the poblem / scientific annotated diagam
The sting tension F T and the gavitational foce F G povide the centipetal foce cicula path. FC equied fo the ball to move in the F F F F C G T C Note: in this example the speed of the ball changes, howeve, it is still tue that a foce equal to the centipetal foce is needed to hold the ball in its cicula obit. When the ball is at point A, the centipetal foce is the sum of the sting tension and the gavitational foce (weight of ball). F F F F CA TA G TA m g v FTA m g The minimum speed v A of the ball to keep moving in a cicula path is when the sting just goes limp FTA 0. F va TA A m g 0 g 9.81.3 m.s 4.68 m.s -1-1
At the bottom of the swing at point B v B v 9.35 m.s A -1 F F F F CB TB G TB TB B m g v B FTB m g F B 9.35 0.5 9.81 N 1.3 N.3 Since the ball s speed changes as it moves aound the cicle, a net toque must act on the ball due to the sting tension and the gavitational foce. Note: The equation given in the Physics Stage 6 Syllabus fo the toque is incoect F you simply cannot multiple two vectos togethe VISUAL PHYSICS ONLINE If you have any feedback, comments, suggestions o coections please email Ian Coope ian.coope@sydney.edu.au Ian Coope School of Physics Univesity of Sydney