A TREATISE ON ELEMENTARY DYNAMICS

By D. ALZIR

CONTENTS DYNAMICS OF A PARTICLE 1 FUNDAMENTAL DEFINITIONS AND PRINCIPLES 1 2 MOTION IN A STRAIGHT LINE 13 3 UNIPLANAR MOTION WHERE THE ACCELERATIONS PARALLEL TO FIXED AXES ARE GIVEN 45 4 UNIPLANAR MOTION REFERRED TO POLAR COORDINATES CENTRAL FORCES 63 5 UNIPLANAR MOTION WHEN THE ACCELERATION IS CENTRAL AND VARYING AS THE INVERSE SQUARE OF THE DISTANCE 107 6 TANGENTIAL AND NORMAL ACCELERATIONS UNIPLANAR CONSTRAINED MOTION 137 7 MOTION IN A RESISTING MEDIUM MOTION OF PARTICLES OF VARYING MASS 171 8 OSCILLATORY MOTION AND SMALL OSCILLATIONS 197 9 MOTION IN THREE DIMENSIONS 223 10 MISCELLANEOUS THE HODOGRAPH MOTION ON REVOLVING CURVES IMPULSIVE TENSIONS OF STRINGS 245 11 MISCELLANEOUS EXAMPLES I 267 CHAPTER 1 FUNDAMENTAL DEFINITIONS AND PRINCIPLES 1 THE VELOCITY OF A POINT IS THE RATE OF ITS DISPLACEMENT SO THAT IF P BE ITS POSITION AT TIME T AND Q THAT AT TIME T T THE LIMITING VALUE OF THE QUANTITY PQ T AS T IS MADE VERY SMALL IS ITS VELOCITY SINCE A DISPLACEMENT HAS BOTH MAGNITUDE AND DIRECTION THE VE LOCITY POSSESSES BOTH ALSO THE LATTER CAN THEREFORE BE REPRESENTED IN MAGNITUDE AND DIRECTION BY A STRAIGHT LINE AND IS HENCE CALLED A VEC TOR QUANTITY 2 A POINT MAY HAVE TWO VELOCITIES IN DIFFERENT DIRECTIONS AT THE SAME INSTANT THEY MAY BE COMPOUNDED INTO ONE VELOCITY BY THE FOLLOWING THEOREM KNOWN AS THE PARALLELOGRAM OF VELOCITIES IF A MOVING POINT POSSESS SIMULTANEOUSLY VELOCITIES WHICH ARE REP RESENTED IN MAGNITUDE AND DIRECTION BY THE TWO SIDES OF A PARALLELO GRAM DRAWN FROM A POINT THEY ARE EQUIVALENT TO A VELOCITY WHICH IS REPRESENTED IN MAGNITUDE AND DIRECTION BY THE DIAGONAL OF THE PAR ALLELOGRAM PASSING THROUGH THE POINT THUS TWO COMPONENT VELOCITIES AB AC ARE EQUIVALENT TO THE RESUL TANT VELOCITY AD WHERE AD IS THE DIAGONAL OF THE PARALLELOGRAM OF WHICH AB AC ARE ADJACENT SIDES 1 1 IF BAC BE A RIGHT ANGLE AND BAD THEN AB ADCOS AC ADSIN AND A VELOCITY V ALONG AD IS EQUIVALENT TO THE TWO COMPONENT VELOCITIES VCOS ALONG AB AND VSIN ALONG AC TRIANGLE OF VELOCITIES IF A POINT POSSESS TWO VELOCITIES COMPLETELY REPRESENTED I E REPRESENTED IN MAGNITUDE DIRECTION AND SENSE BY TWO STRAIGHT LINES AB AND BC THEIR RESULTANT IS COMPLETELY REPRE SENTED BY AC FOR COMPLETING THE PARALLELOGRAM ABCD THE VELOCI TIES AB BC ARE EQUIVALENT TO AB AD WHOSE RESULTANT IS AC PARALLELEPIPED OF VELOCITIES IF A POINT POSSESS THREE VELOCITIES COMPLETELY REPRESENTED BY THREE STRAIGHT LINES OA OB OC THEIR RE SULTANT IS BY SUCCESSIVE APPLICATIONS OF THE PARALLELOGRAM OF VE LOCITIES COMPLETELY REPRESENTED BY OD THE DIAGONAL OF THE PARAL LELEPIPED OF WHICH OA OB OC ARE CONTERMINOUS EDGES SIMILARLY OA OB AND OC ARE THE COMPONENT VELOCITIES OF OD IF OA OB AND OC ARE MUTUALLY AT RIGHT ANGLES AND U V W ARE THE VELOCITIES OF THE MOVING POINT ALONG THESE DIRECTIONS THE RESULTANT VELOCITY IS U2 V2 W2 ALONG A LINE WHOSE DIRECTION COSINES ARE PROPORTIONAL TO W V W AND ARE THUS EQUAL TO U U2 V2 W2 V U2 V2 W2 AND W U2 V2 W2 SIMILARLY IF OD BE A STRAIGHT LINE WHOSE DIRECTION COSINES RE FERRED TO THREE MUTUALLY PERPENDICULAR LINES OA OB OC ARE L M N THEN A VELOCITY V ALONG OD IS EQUIVALENT TO COMPONENT VELOCITIES LV MV NV ALONG OA OB AND OC RESPECTIVELY 3 CHANGE OF VELOCITY ACCELERATION IF AT ANY INSTANT THE VELOC ITY OF A MOVING POINT BE REPRESENTED BY OA AND AT ANY SUBSEQUENT INSTANT BY OB AND IF THE PARALLELOGRAM OABC BE COMPLETED WHOSE A TREATISE ON ELEMENTARY DYNAMICS USE OF COLLEGES AND SCHOOLS 2 DIAGONAL IS OB THEN OC OR AB REPRESENTS THE VELOCITY WHICH MUST BE COMPOUNDED WITH OA TO GIVE OB I E IT IS THE CHANGE IN THE VELOCITY OF THE MOVING POINT ACCELERATION IS THE RATE OF CHANGE OF VELOCITY I E IF OA OB REPRE SENT THE VELOCITIES AT TIMES T AND T T THEN THE LIMITING VALUE OF BA T I E THE LIMITING VALUE OF THE RATIO OF THE CHANGE IN THE VELOCITY TO THE CHANGE IN THE TIME AS T BECOMES INDE NITELY SMALL IS THE ACCEL ERATION OF THE MOVING POINT AS IN THE CASE OF VELOCITIES A MOVING POINT MAY POSSESS SIMULTANEOUSLY ACCELERATIONS IN DIFFERENT DIREC TIONS AND THEY MAY BE COMPOUNDED INTO ONE BY A THEOREM KNOWN AS THE PARALLELOGRAM OF ACCELERATIONS SIMILAR TO THE PARALLELOGRAM OF VELOCITIES AS ALSO IN THE CASE OF VELOCITIES AN ACCELERATION MAY BE RESOLVED INTO TWO COMPONENT ACCELERATIONS THE RESULTS OF ART 2 ARE ALSO TRUE FOR ACCELERATIONS AS WELL AS VE LOCITIES 4 RELATIVE VELOCITY WHEN THE DISTANCE BETWEEN TWO MOVING POINTS IS ALTERING EITHER IN DIRECTION OR IN MAGNITUDE OR IN BOTH EACH POINT IS SAID TO HAVE A VELOCITY RELATIVE TO THE OTHER P R Q A B SUPPOSE THE VELOCITIES OF TWO MOVING POINTS A AND B TO BE REPRE SENTED BY THE TWO LINES AP AND BQ WHICH ARE NOT NECESSARILY IN THE A TREATISE ON ELEMENTARY DYNAMICS USE OF COLLEGES AND SCHOOLS 3