DYNAMICS OF ROTATION
CONTENTS 1 STRATEGIES FOR SOLVING PROBLEMS 1 1 GENERAL STRATEGIES 1 2 UNITS, DIMENSIONAL ANALYSIS 1 3 APPROXIMATIONS, LIMITING CASES 1 4 SOLVING DIFFERENTIAL EQUATIONS NUMERICALLY 1 5 PROBLEMS 1 6 EXERCISES 1 7 SOLUTIONS 2 STATICS 2 1 BALANCING FORCES 2 2 BALANCING TORQUES 2 3 PROBLEMS 2 4 EXERCISES 2 5 SOLUTIONS 3 USING F = MA 3 1 NEWTON’S LAWS 3 2 FREE-BODY DIAGRAMS 3 3 SOLVING DIFFERENTIAL EQUATIONS 3 4 PROJECTILE MOTION 3 5 MOTION IN A PLANE, POLAR COORDINATES 3 6 PROBLEMS 3 7 EXERCISES 3 8 SOLUTIONS 4 OSCILLATIONS 4 1 LINEAR DIFFERENTIAL EQUATIONS 4 2 SIMPLE HARMONIC MOTION DAMPED HARMONIC MOTION DRIVEN (AND DAMPED) HARMONIC MOTION COUPLED OSCILLATORS PROBLEMS EXERCISES SOLUTIONS 5 CONSERVATION OF ENERGY AND MOMENTUM CONSERVATION OF ENERGY IN ONE DIMENSION SMALL OSCILLATIONS CONSERVATION OF ENERGY IN THREE DIMENSIONS GRAVITY MOMENTUM THE CENTER OF MASS FRAME COLLISIONS INHERENTLY INELASTIC PROCESSES 5 10 EXERCISES 5 11 SOLUTIONS 6 THE LAGRANGIAN METHOD THE EULER–LAGRANGE EQUATIONS THE PRINCIPLE OF STATIONARY ACTION FORCES OF CONSTRAINT CHANGE OF COORDINATES CONSERVATION LAWS NOETHER’S THEOREM OTHER APPLICATIONS 6 10 EXERCISES 6 11 SOLUTIONS 7 CENTRAL FORCES CONSERVATION OF ANGULAR MOMENTUM THE EFFECTIVE POTENTIAL SOLVING THE EQUATIONS OF MOTION GRAVITY, KEPLER’S LAWS 8 ANGULAR MOMENTUM, PART I (CONSTANT ˆL) PANCAKE OBJECT IN X-Y PLANE NONPLANAR OBJECTS CALCULATING MOMENTS OF INERTIA TORQUE ANGULAR IMPULSE 9 ANGULAR MOMENTUM, PART II (GENERAL ˆL) PRELIMINARIES CONCERNING ROTATIONS THE INERTIA TENSOR PRINCIPAL AXES TWO BASIC TYPES OF PROBLEMS EULER’S EQUATIONS FREE SYMMETRIC TOP HEAVY SYMMETRIC TOP 10 ACCELERATING FRAMES OF REFERENCE RELATING THE COORDINATES THE FICTITIOUS FORCES TIDES 11 RELATIVITY (KINEMATICS) MOTIVATION THE POSTULATES THE FUNDAMENTAL EFFECTS THE LORENTZ TRANSFORMATIONS VELOCITY ADDITION THE INVARIANT INTERVAL MINKOWSKI DIAGRAMS THE DOPPLER EFFECT RAPIDITY 11 10 RELATIVITY WITHOUT C 11 11 PROBLEMS 11 12 EXERCISES 11 13 SOLUTIONS 12 RELATIVITY (DYNAMICS) ENERGY AND MOMENTUM TRANSFORMATIONS OF E AND P COLLISIONS AND DECAYS PARTICLE-PHYSICS UNITS FORCE ROCKET MOTION RELATIVISTIC STRINGS 12 10 SOLUTIONS 13 4-VECTORS DEFINITION OF 4-VECTORS EXAMPLES OF 4-VECTORS PROPERTIES OF 4-VECTORS ENERGY, MOMENTUM FORCE AND ACCELERATION THE FORM OF PHYSICAL LAWS 14 GENERAL RELATIVITY THE EQUIVALENCE PRINCIPLE TIME DILATION UNIFORMLY ACCELERATING FRAME MAXIMAL-PROPER-TIME PRINCIPLE TWIN PARADOX REVISITED CHAPTER GENERAL STRATEGIES THERE ARE A NUMBER OF GENERAL STRATEGIES YOU SHOULD INVOKE WITHOUT HESITATION WHEN SOLVING A PROBLEM THEY ARE: 1 DRAW A DIAGRAM, IF APPROPRIATE IN THE DIAGRAM, BE SURE TO LABEL CLEARLY ALL THE RELEVANT QUANTITIES (FORCES, LENGTHS, MASSES, ETC ) DIAGRAMS ARE ABSOLUTELY CRITICAL IN CERTAIN TYPES OF PROBLEMS FOR EXAMPLE, IN PROBLEMS INVOLVING “FREE-BODY” DIAGRAMS (DISCUSSED IN CHAPTER 3) OR RELATIVISTIC KINEMATICS (DISCUSSED IN CHAPTER 11), DRAWING A DIAGRAM CAN CHANGE A HOPELESSLY COMPLICATED PROBLEM INTO A NEAR-TRIVIAL ONE AND EVEN IN CASES WHERE DIAGRAMS AREN’T THIS CRUCIAL, THEY’RE INVARIABLY VERY HELPFUL A PICTURE IS DEFINITELY WORTH A THOUSAND WORDS (AND EVEN A FEW MORE, IF YOU LABEL THINGS!) 2 WRITE DOWN WHAT YOU KNOW, AND WHAT YOU ARE TRYING TO FIND IN A SIMPLE PROBLEM, YOU MAY JUST DO THIS IN YOUR HEAD WITHOUT REALIZING IT BUT IN MORE DIFFICULT PROBLEMS, IT IS VERY USEFUL TO EXPLICITLY WRITE THINGS OUT FOR EXAMPLE, IF THERE ARE THREE UNKNOWNS THAT YOU’RE TRYING TO FIND, BUT YOU’VE WRITTEN DOWN ONLY TWO FACTS, THEN YOU KNOW THERE MUST BE ANOTHER FACT YOU’RE MISSING (ASSUMING THAT THE PROBLEM IS IN FACT SOLVABLE), SO YOU CAN GO SEARCHING FOR IT IT MIGHT BE A CONSERVATION LAW, OR AN F = MA EQUATION, ETC 3 SOLVE THINGS SYMBOLICALLY IF YOU ARE SOLVING A PROBLEM WHERE THE GIVEN QUANTITIES ARE SPECIFIED NUMERICALLY, YOU SHOULD IMMEDIATELY CHANGE THE NUMBERS TO LETTERS AND SOLVE THE PROBLEM IN TERMS OF THE LETTERS AFTER YOU OBTAIN AN ANSWER IN TERMS OF THE LETTERS, YOU CAN PLUG IN THE ACTUAL NUMERICAL VALUES TO OBTAIN A NUMERICAL ANSWER THERE ARE MANY ADVANTAGES TO USING LETTERS: IT’S QUICKER IT’S MUCH EASIER TO MULTIPLY A G BY AN ?BY WRITING THEM DOWN ON A PIECE OF PAPER NEXT TO EACH OTHER, THAN IT IS TO MULTIPLY THEM TOGETHER ON A CALCULATOR AND WITH THE LATTER STRATEGY, YOU’D UNDOUBTEDLY HAVE TO PICK UP YOUR CALCULATOR AT LEAST A FEW TIMES DURING THE COURSE OF A PROBLEM YOU’RE LESS LIKELY TO MAKE A MISTAKE IT’S VERY EASY TO MISTYPE AN 8 FOR A 9 IN A CALCULATOR, BUT YOU’RE PROBABLY NOT GOING TO MISWRITE A Q FOR A G ON A PIECE OF PAPER BUT IF YOU DO, YOU’LL QUICKLY REALIZE THAT IT SHOULD BE A G YOU CERTAINLY WON’T JUST GIVE UP ON THE PROBLEM AND DEEM IT UNSOLVABLE BECAUSE NO ONE GAVE YOU THE VALUE OF Q! YOU CAN DO THE PROBLEM ONCE AND FOR ALL IF SOMEONE COMES ALONG AND SAYS, OOPS, THE VALUE OF ?IS ACTUALLY 2 4 M INSTEAD OF 2 3 M, THEN YOU WON’T HAVE TO DO THE WHOLE PROBLEM AGAIN YOU CAN SIMPLY PLUG THE NEW VALUE OF ?INTO YOUR FINAL SYMBOLIC ANSWER YOU CAN SEE THE GENERAL DEPENDENCE OF YOUR ANSWER ON THE VARIOUS GIVEN QUAN- TITIES FOREXAMPLE, YOUCANSEETHATITGROWSWITHQUANTITIESAANDB, DECREASESWITH C, AND DOESN’T DEPEND ON D THERE IS MUCH, MUCH MORE INFORMATION CONTAINED IN A SYMBOLIC ANSWER THAN IN A NUMERICAL ONE AND BESIDES, SYMBOLIC ANSWERS NEARLY ALWAYS LOOK NICE AND PRETTY YOU CAN CHECK UNITS AND SPECIAL CASES THESE CHECKS GO HAND-IN-HAND WITH THE PREVIOUS “GENERAL DEPENDENCE” ADVANTAGE BUT SINCE THEY’RE SO IMPORTANT, WE’LL POSTPONE THEIR DISCUSSION AND DEVOTE SECTIONS 1 2 AND 1 3 TO THEM HAVING SAID ALL THIS, IT SHOULD BE NOTED THAT THERE ARE OCCASIONALLY TIMES WHEN THINGS GET A BIT MESSY WHEN WORKING WITH LETTERS FOR EXAMPLE, SOLVING A SYSTEM OF THREE EQUATIONS IN THREE UNKNOWNS MIGHT BE RATHER CUMBERSOME UNLESS YOU PLUG IN THE ACTUAL NUMBERS BUT IN THE VAST MAJORITY OF PROBLEMS, IT IS HIGHLY ADVANTAGEOUS TO WORK ENTIRELY WITH LETTERS 4 CONSIDER UNITS/DIMENSIONS THIS IS EXTREMELY IMPORTANT SEE SECTION 1 2 FOR A DETAILED DISCUSSION DYNAMICS OF ROTATION 5 CHECK LIMITING/SPECIAL CASES THIS IS ALSO EXTREMELY IMPORTANT SEE SECTION 1 3 FOR A DETAILED DISCUSSION 6 CHECK ORDER OF MAGNITUDE IF YOU END UP GETTING A NUMERICAL ANSWER IF YOU END UP WITH AN ACTUAL NUMERICAL ANSWER TO A PROBLEM, BE SURE TO DO A SAN- ITY CHECK TO SEE IF THE NUMBER IS REASONABLE IF YOU’VE CALCULATED THE DISTANCE ALONG THE GROUND THAT A CAR SKIDS BEFORE IT COMES TO REST, AND IF YOU’VE GOTTEN AN ANSWER OF A KILOMETER OR A MILLIMETER, THEN YOU KNOW YOU’VE PROBABLY DONE SOMETHING WRONG ERRORS OF THIS SORT OFTEN COME FROM FORGETTING SOME POWERS OF 10 (SAY, WHEN CONVERTING KILOMETERS TO METERS) OR FROM MULTIPLYING SOMETHING INSTEAD OF DIVIDING (ALTHOUGH YOU SHOULD BE ABLE TO CATCH THIS BY CHECKING YOUR UNITS, TOO) YOU WILL INEVITABLY ENCOUNTER PROBLEMS, PHYSICS ONES OR OTHERWISE, WHERE YOU DON’T END UP OBTAINING A RIGOROUS ANSWER, EITHER BECAUSE THE CALCULATION IS INTRACTABLE, OR BECAUSE YOU JUST DON’T FEEL LIKE DOING IT BUT IN THESE CASES IT’S USUALLY STILL POSSIBLE TO MAKE AN EDUCATED GUESS, TO THE NEAREST POWER OF 10 FOR EXAMPLE, IF YOU WALK PAST A BUILDING AND HAPPEN TO WONDER HOW MANY BRICKS ARE IN IT, OR WHAT THE LABOR COST WAS IN CONSTRUCTING IT, THEN YOU CAN PROBABLY GIVE A REASONABLE ANSWER WITHOUT DOING ANY SEVERE COMPUTATIONS THE PHYSICIST ENRICO FERMI WAS KNOWN FOR HIS ABILITY TO ESTIMATE THINGS QUICKLY AND PRODUCE ORDER-OF-MAGNITUDE GUESSES WITH ONLY MINIMAL CALCULATION HENCE, A PROBLEM WHERE THE GOAL IS TO SIMPLY OBTAIN THE NEAREST POWER-OF-10 ESTIMATE IS KNOWN AS A “FERMI PROBLEM ” OF COURSE, SOMETIMES IN LIFE YOU NEED TO KNOW THINGS TO BETTER ACCURACY THAN THE NEAREST POWER OF 10 HOW FERMI COULD ESTIMATE THINGS! LIKE THE WELL-KNOWN OLYMPIC TEN RINGS, AND THE ONE HUNDRED STATES, AND WEEKS WITH TEN DATES, AND BIRDS THAT ALL FLY WITH ONE WINGS IN THE FOLLOWING TWO SECTIONS, WE’LL DISCUSS THE VERY IMPORTANT STRATEGIES OF CHECKING UNITS AND SPECIAL CASES THEN IN SECTION 1 4 WE’LL DISCUSS THE TECHNIQUE OF SOLVING PROBLEMS NUMERICALLY, WHICH IS WHAT YOU NEED TO DO WHEN YOU END UP WITH A SET OF EQUATIONS YOU CAN’T FIGURE OUT HOW TO SOLVE SECTION 1 4 ISN’T QUITE ANALOGOUS TO SECTIONS 1 2 AND 1 3, IN THAT THESE FIRST TWO ARE RELEVANT TO BASICALLY ANY PROBLEM YOU’LL EVER DO, WHEREAS SOLVING EQUATIONS NUMERICALLY IS SOMETHING YOU’LL DO ONLY FOR OCCASIONAL PROBLEMS BUT IT’S NEVERTHELESS SOMETHING THAT EVERY PHYSICS STUDENT SHOULD KNOW IN ALL THREE OF THESE SECTIONS, WE’LL INVOKE VARIOUS RESULTS DERIVED LATER IN THE BOOK FOR THE PRESENT PURPOSES, THE DERIVATIONS OF THESE RESULTS ARE COMPLETELY IRRELEVANT, SO DON’T WORRY AT ALL ABOUT THE PHYSICS BEHIND THEM – THERE WILL BE SOLUTION: THE ONLY DIMENSIONFUL QUANTITIES GIVEN IN THE PROBLEM ARE [M] = M, [?] = L, AND [G] = L/T2 BUT THERE IS ONE MORE QUANTITY, THE MAXIMUM ANGLE ?0, WHICH IS DIMENSIONLESS (AND EASY TO FORGET) OUR GOAL IS TO FIND THE FREQUENCY, WHICH 1 WHEN YOU CHECK UNITS AT THE END OF A CALCULATION, YOU WILL INVARIABLY BE WORKING WITH THE KG,M,S NOTATION SO THAT NOTATION WILL INEVITABLY GET USED MORE BUT I’LL USE THE M, L, T NOTATION HERE, BECAUSE I THINK IT’S A LITTLE MORE INSTRUCTIVE AT ANY RATE, JUST REMEMBER THAT THE LETTER M (OR M) STANDS FOR “METER” IN ONE CASE, AND “MASS” IN THE OTHER 0 /16 + 2 SINCE THERE IS ONLY ONE MASS IN THE PROBLEM, THERE IS NO WAY THAT THE FREQUENCY (WITH UNITS OF 1/T) CAN DEPEND ON [M] = M IF IT DID, THERE WOULD BE NOTHING TO CANCEL THE UNITS OF MASS AND PRODUCE A PURE INVERSE-TIME 3 WE CLAIMED ABOVE THAT THE ONLY COMBINATION OF OUR GIVEN DIMENSIONFUL QUANTITIES THAT HAS UNITS OF 1/T IS ?G/? THIS IS EASY TO SEE HERE, BUT IN MORE COMPLICATED PROBLEMS WHERE THE CORRECT COMBINATION ISN’T SO OBVIOUS, THE FOLLOWING METHOD WILL ALWAYS WORK WRITE DOWN A GENERAL PRODUCT OF THE GIVEN DIMENSIONFUL QUANTITIES RAISED TO ARBITRARY POWERS (MA?BGC IN THIS PROBLEM), AND THEN WRITE OUT THE UNITS OF THIS PRODUCT IN TERMS OF A, B, AND C IF WE WANT TO OBTAIN UNITS OF 1/T HERE, THEN WE NEED M ALB L T C = 0 /24 + SO IN CONTRAST WITH THE FREQUENCY RESULT ABOVE, THE MAXIMUM ANGLE ?0 PLAYS A CRITICAL ROLE IN THE ENERGY 2 WE’LL MEASURE FREQUENCY HERE IN RADIANS PER SECOND, DENOTED BY ? SO WE’RE ACTUALLY TALKING ABOUT THE “ANGULAR FREQUENCY ” JUST DIVIDE BY 2? (WHICH DOESN’T AFFECT THE UNITS) TO OBTAIN THE “REGULAR” FREQUENCY IN CYCLES PER SECOND (HERTZ), USUALLY DENOTED BY ? WE’LL TALK AT GREAT LENGTH ABOUT OSCILLATIONS IN CHAPTER 4 DEPENDENCE IN THIS CASE THIS ANSWER MUST REDUCE TO C K/M FOR B = 0 HENCE, F MUST BE OF THE FORM F (Y) = C + C1Y + C2Y2 + ? EXAMPLE (LOW-ORBIT SATELLITE): A SATELLITE OF MASS M TRAVELS IN A CIRCULAR ORBIT JUST ABOVE THE EARTH’S SURFACE WHAT CAN WE SAY ABOUT ITS SPEED? SOLUTION: THE ONLY DIMENSIONFUL QUANTITIES IN THE PROBLEM ARE [M] = M, [G] = L/T2, AND THE RADIUS OF THE EARTH [R] = L 3 OUR GOAL IS TO FIND THE SPEED, WHICH HAS UNITS OF L/T THE ONLY COMBINATION OF OUR DIMENSIONFUL QUANTITIES WITH THESE UNITS IS V = C GR (1 5) IT TURNS OUT THAT C = 1 3 YOU MIGHT ARGUE THAT THE MASS OF THE EARTH, ME, AND NEWTON’S GRAVITATIONAL CONSTANT, G, SHOULD BE ALSO INCLUDED HERE, BECAUSE NEWTON’S GRAVITATIONAL FORCE LAW FOR A PARTICLE ON THE SURFACE OF THE EARTH IS F = GMEM/R2 BUT SINCE THIS FORCE CAN BE WRITTEN AS M(GME/R2) ?MG, WE CAN ABSORB THE EFFECTS OF ME AND G INTO G ? 1 ?E??T (1 6) THESE EXPRESSIONS ARE A BIT COMPLICATED, SO FOR ALL YOU KNOW, I COULD HAVE MADE A TYPO IN WRITING THEM DOWN OR WORSE, I COULD HAVE COMPLETELY BOTCHED THE SOLU- TION SO LET’S LOOK AT SOME LIMITING CASES IF THESE LIMITING CASES YIELD EXPECTED RESULTS, THEN WE CAN FEEL A LITTLE MORE CONFIDENT THAT THE ANSWERS ARE ACTUALLY CORRECT IF T IS VERY SMALL (MORE PRECISELY, IF ?T ?1; SEE THE DISCUSSION FOLLOWING THIS EXAM- PLE), THEN WE CAN USE THE TAYLOR SERIES, E?X ?1 ?X + X2/2, TO MAKE APPROXIMATIONS 2 , (1 8) PLUS TERMS OF HIGHER ORDER IN ?T AGAIN, THIS ANSWER IS EXPECTED, BECAUSE WE ESSENTIALLY HAVE A FREELY FALLING BODY AT THE START, SO THE DISTANCE FALLEN IS THE STANDARD GT2/2 WE CAN ALSO LOOK AT LARGE T (OR RATHER, LARGE ?T) IN THIS CASE, E??T IS ESSENTIALLY ZERO, SO THE V(T) IN EQ (1 6) BECOMES (THERE’S NO NEED FOR A TAYLOR SERIES IN THIS CASE) V(T) ??G ? (1 9) THIS IS THE “TERMINAL VELOCITY ” ITS VALUE MAKES SENSE, BECAUSE IT IS THE VELOCITY FOR WHICH THE TOTAL FORCE, ?MG ?M?V, VANISHES FOR LARGE T, EQ (1 6) ALSO GIVES Y(T) ?H ?GT ? + G ?2 (1 10) APPARENTLY FOR LARGE T, G/?2 IS THE DISTANCE (AND THIS DOES INDEED HAVE UNITS OF LENGTH, BECAUSE ? HAS UNITS OF T?1, BECAUSE M?V HAS UNITS OF FORCE) THAT OUR BALL LAGS BEHIND ANOTHER BALL THAT STARTED OUT ALREADY AT THE TERMINAL VELOCITY, ?G/? WHENEVER YOU DERIVE APPROXIMATE ANSWERS AS WE JUST DID, YOU GAIN SOMETHING AND YOU LOSE SOMETHING YOU LOSE SOME TRUTH, OF COURSE, BECAUSE YOUR NEW ANSWER IS TECHNICALLY NOT CORRECT BUT YOU GAIN SOME AESTHETICS YOUR NEW ANSWER IS INVARIABLY MUCH CLEANER (SOMETIMES INVOLVING ONLY ONE TERM), AND THIS MAKES IT A LOT EASIER TO SEE WHAT’S GOING ON IN THE ABOVE EXAMPLE, IT ACTUALLY MAKES NO SENSE TO LOOK AT THE LIMIT WHERE T IS SMALL OR LARGE, BECAUSE T HAS DIMENSIONS IS A YEAR A LARGE OR SMALL TIME? HOW ABOUT A HUNDREDTH OF A SECOND? THERE IS NO WAY TO ANSWER THIS WITHOUT KNOWING WHAT PROBLEM YOU’RE DEALING WITH A YEAR IS SHORT ON THE TIME SCALE OF GALACTIC EVOLUTION, BUT A HUNDREDTH OF A SECOND IS LONG ON THE TIME SCALE OF A NUCLEAR PROCESS IT MAKES SENSE ONLY TO LOOK AT THE LIMIT OF A SMALL (OR LARGE) DIMENSIONLESS QUANTITY IN THE ABOVE EXAMPLE, THIS QUANTITY IS ?T THE GIVEN CONSTANT ? HAS UNITS OF T ?1, SO 1/? SETS A TYPICAL TIME SCALE FOR THE SYSTEM IT