FUNDAMENTALS OF PHYSICS
CONTENT 1 CIRCULAR MOTION PROBLEMS MULTIPLE CHOICE ITEMS MARKSCHEME 2 UNIVERSAL GRAVITATION 3 ELECTROSTATICS ELECTRIC FORCE ELECTRIC FIELD ELECTRIC POTENTIAL AND POTENTIAL ENERGY 4 MAGNETISM RIGHT HAND RULE EXERCISES MAGNETIC FIELD AROUND CURRENT CARRYING WIRE MAGNETIC FORCE ON MOVING CHARGES 5 MAGNETIC INDUCTION 6 ALTERNATING CURRENT CIRCULAR MOTION Y X Q P VB 3R 2R VA O PROBLEMS 1 TWO BODIES M1 AND M2 ,SHOWN IN FIGURE ON RIGHT, ARE CIRCLING ON THE SAME PLANE WITH THE PERIODS OF T1=3S AND T2=5S SHOW THEIR POSITIONS AFTER 12S ON THE FIGURE ON LEFT 2 AN OBJECT UNDERGOES UNIFORM CIRCULAR MOTION WITH A PERIOD OF 72 SECONDS FIND THE TIME REQUIRED FOR THE OBJECT TO MOVE FROM P TO Q 3 TWO OBJECTS A AND B ARE ROTATING ON THE CIRCULAR ORBITS ON THE SAME PLANE AS SHOWN IF THE PERIODS OF A AND B ARE 2 SEC AND 3 SECONDS RESPECTIVELY, WHAT IS THE RATIO OF VA/VB? 4 A MASS OF 1 5 KG MOVES IN A HORIZONTAL CIRCLE OF RADIUS 0 25M AT 2 REV/S CALCULATE THE MAGNITUDE OF; (A) THE LINEAR SPEED (B) THE ACCELERATION (C) CENTRIPETAL FORCE ACTING ON THE BODY COLLEGE PHYSICS FOR COLLEGE STUDENTS AND M2 ARE EQUAL, WHAT IS THE RATIO OF M1/ M2? 7 TWO OBJECTS ARE ATTACHED TO THE MIDPOINT AND TO THE END OF A ROPE AND ARE BEING TURNED AROUND HORIZONTALLY AS SHOWN IN FIGURE BELOW THE MASS OF INNER OBJECT IS TWICE OF THE MASS OF OUTER ONE (A) WHICH ONE OF THE FOLLOWINGS IS EQUAL FOR THESE MASSES? I PERIOD II LINEAR VELOCITY III ANGULAR VELOCITY IV CENTRIPETAL ACCELERATION (B) WHAT IS THE RATIO OF TENSIONS IN THE ROPES OA AND AB? COLLEGE PHYSICS FOR COLLEGE STUDENTS V C T P V C BOTTOM TOP SIDE 8 AN OBJECT OF MASS 0 5 KG IS ATTACHED TO THE END OF A ROPE OF 1 M IT PERFORMS UNIFORM CIRCULAR MOTION IN VERTICAL PLANE IF, IT MAKES 2 REVOLUTIONS PER SECOND; (A) FIND THE LINEAR VELOCITY OF THE OBJECT (B) FIND THE TENSION IN THE STRING AT THE TOP, BOTTOM AND SIDE POINTS OF THE CIRCULAR PATH 9 MASS M IS ATTACHED TO ONE END OF A ROPE OF LENGTH 1 M AND IS ROTATED AROUND CENTRE C, ALONG A VERTICALLY ALIGNED CIRCULAR PATH WHEN MASS M IS AT THE LOWEST POINT ON ITS PATH, THE TENSION IN THE ROPE IS MEASURED TO BE “3MG” WHAT IS THE VELOCITY OF THE MASS M? 10 A CIRCULAR HORIZONTAL PLATFORM ROTATES AROUND ITS CENTRAL AXIS WITH A CONSTANT ANGULAR SPEED A 0 2 KG MASS IS PLACED AT THE EDGE OF THE PLATFORM AS SHOWN WHAT IS THE MAXIMUM VALUE OF THE LINEAR VELOCITY THE PLATFORM CAN HAVE, TO KEEP MASS ON IT, IF THE COEFFICIENT OF FRICTION BETWEEN THEM IS 0 5? 0 8 M M COLLEGE PHYSICS FOR COLLEGE STUDENTS L B A 11 A SMALL CART IS RELEASED DOWN A TRACK, THE BOTTOM PART OF WHICH HAS BEEN BENT INTO A LOOP AS SHOWN IN FIGURE BELOW WHAT MUST BE THE MINIMUM INITIAL HEIGHT THAT THE CART IS RELEASED SO THAT IT CAN PASS THROUGH THE HIGHEST POINT OF LOOP (POINT L) WITHOUT FALLING DOWN? EXPRESS THE HEIGHT (H) IN TERMS OF RADIUS OF CIRCLE (R) 12 AN OBJECT IS RELEASED FROM POINT A OF THE FRICTIONLESS SURFACE WHAT IS THE REACTION FORCE EXERTED BY THE RAIL TO THE OBJECT AT POINT B? 13 AN OBJECT OF MASS, M=1 KG IS LET TO SWING FROM POINT B IT IS ATTACHED TO A ROPE OF LENGTH, L NEGLECTING THE FRICTIONAL EFFECTS, WHAT IS THE TENSION IN THE STRING AT POINT A? H R K L R 2R 30? B A COLLEGE PHYSICS FOR COLLEGE STUDENTS M1=2KG M2=1KG R=2M V=? K 14 THE OBJECT IN FIGURE IS RELEASED TO SWING FROM POINT A IT PASSES THROUGH POINT B WITH A VELOCITY OF V AND PASSES THROUGH POINT C WITH A VELOCITY OF V/2 IT IS KNOWN THAT THE MASS OF OBJECT IS 0 1KG AND THE LENGTH OF STRING IS 4M (A) WHAT IS THE INITIAL HEIGHT (H) OF OBJECT AT POINT A? (FRICTIONAL EFFECTS ARE NEGLECTED ) (B) WHAT IS THE TENSION IN THE STRING AT POINT A, 15 THE OBJECTS WITH MASSES M1=2KG AND M2=1KG TIGHT TOGETHER, ARE IN EQUILIBRIUM AS IN FIGURE IF THE SURFACE IS FRICTIONLESS, FIND THE VELOCITY WITH WHICH OBJECT K MOVE? 16 AN OBJECT IS RELEASED FROM THE TOP OF A SEMICIRCULAR SURFACE TO SLIDE DOWN AT WHICH HEIGHT, THE OBJECT WOULD LEAVE THE SURFACE? R H=? 2 4M H A B C COLLEGE PHYSICS FOR COLLEGE STUDENTS 17 FIGURE BELOW SHOWS THE LONDON EYE (A) IT HAS MANY CAPSULES ATTACHED AROUND THE EDGE OF A LARGE VERTICAL WHEEL OF RADIUS 60M THE WHEEL ROTATES ABOUT A HORIZONTAL AXIS SUCH THAT EACH CAPSULE HAS A CONSTANT SPEED OF 0 26MS-1 (I) CALCULATE THE TIME TAKEN FOR THE WHEEL TO MAKE ONE COMPLETE ROTAION (II) EACH CAPSULE HAS A MASS OF 9 7X103KG CALCULATE THE CENTRIPETAL FORCE WHICH MUST ACT ON THE CAPSULE TO MAKE IT ROTATE WITH THE WHEEL (B) DETERMINE THE DIRECTION AND MAGNITUDE OF FORCE EXERTED BY THE WHEEL ON THE CAPSULE, WHEN THE CAPSULE IS (I) AT THE LOWEST POSITION ON THE WHEEL (II) AT THE HEIGHEST POSITION ON THE WHEEL COLLEGE PHYSICS FOR COLLEGE STUDENTS (C) FIGURE BELOW SHOWS THE DRUM OF A SPIN-DRYER AS IT ROTATES A DRY SOCK S IS SHOWN ON THE INSIDE SURFACE OF THE ROTATING DRUM (I) DRAW ARROWS ON FIGURE TO SHOW THE DIRECTION OF CENTRIPERTAL FORCE ACTING ON S WHEN IT IS AT POINTS A, B AND C (II) STATE AND EXPLAIN AT WHICH POSITION, THE NORMAL FORCE BETWEEN THE SOCK AND THE DRUM WILL BE THE GREATEST: THE LEAST: 18 THIS QUESTION IS ABOUT CIRCULAR MOTION THE DIAGRAM SHOWS A CAR MOVING AT A CONSTANT SPEED OVER A CURVED BRIDGE AT THE POSITION SHOWN, THE TOP SURFACE OF THE BRIDGE HAS A RADIUS OF CURVATURE OF 50 M (A) EXPLAIN WHY THE CAR IS ACCELERATING EVEN THOUGH IT IS MOVING WITH A CONSTANT SPEED [2] (B) ON THE DIAGRAM, DRAW AND LABEL THE VERTICAL FORCES ACTING ON THE CAR IN THE POSITION SHOWN [2] (C) CALCULATE THE MAXIMUM SPEED AT WHICH THE CAR WILL STAY IN CONTACT WITH THE BRIDGE [3] TOP SURFACE OF BRIDGE OF RADIUS OF CURVATURE 50M CAR COLLEGE PHYSICS FOR COLLEGE STUDENTS 19 THIS QUESTION IS ABOUT THE KINEMATICS AND DYNAMICS OF CIRCULAR MOTION (A) A CAR GOES ROUND A CURVE IN A ROAD AT CONSTANT SPEED EXPLAIN WHY, ALTHOUGH ITS SPEED IS CONSTANT, IT IS ACCELERATING (2) IN THE DIAGRAM BELOW, A MARBLE (SMALL GLASS SPHERE) ROLLS DOWN A TRACK, THE BOTTOM PART OF WHICH HAS BEEN BENT INTO A LOOP THE END A OF THE TRACK, FROM WHICH THE MARBLE IS RELEASED, IS AT A HEIGHT OF 0 80 M ABOVE THE GROUND POINT B IS THE LOWEST POINT AND POINT C THE HIGHEST POINT OF THE LOOP THE DIAMETER OF THE LOOP IS 0 35 M THE MASS OF THE MARBLE IS 0 050 KG FRICTION FORCES AND ANY GAIN IN KINETIC ENERGY DUE TO THE ROTATING OF THE MARBLE CAN BE IGNORED THE ACCELERATION DUE TO GRAVITY, G = 10 MS–2 CONSIDER THE MARBLE WHEN IT IS AT POINT C (B) (I) ON THE DIAGRAM OPPOSITE, DRAW AN ARROW TO SHOW THE DIRECTION OF THE RESULTANT FORCE ACTING ON THE MARBLE (1) (II) STATE THE NAMES OF THE TWO FORCES ACTING ON THE MARBLE (2) (III) DEDUCE THAT THE SPEED OF THE MARBLE IS 3 0 MS–1 (3) (IV) DETERMINE THE RESULTANT FORCE ACTING ON THE MARBLE AND HENCE DETERMINE THE REACTION FORCE OF THE TRACK ON THE MARBLE (4) (TOTAL 12 MARKS) A MARBLE GROUND B C 0 80 M 0 35 M COLLEGE PHYSICS FOR COLLEGE STUDENTS 20 BANKED TRACK A CAR TRAVELS AT A STEADY SPEED V IN A CIRCULAR PATH OF RADIUS R ON A CIRCULAR TRACK BANKED AT AN ANGLE ? AS SHOWN IN THE PLAN AND SIDE VIEWS IN THE DIAGRAM THE CAR’S SPEED IS SUCH THAT THERE IS NO SIDEWAYS FRICTIONAL FORCE BETWEEN THE TYRES AND THE TRACK (A) DOES THE CAR HAVE AN ACCELERATION? EXPLAIN WHY OR WHY NOT IF YOU SAY YES, STATE ITS DIRECTION [2] (B) THE CAR ON THE TRACK IS REPRESENTED BY A BLOCK IN THE FIGURE BELOW, MOVING PERPENDICULAR TOTHE PAGE DRAW A FORCE DIAGRAM, SHOWING AND LABELLING ALL THE FORCES ACTING ON THE MOVING CAR [2] (C) IS THERE A RESULTANT FORCE ON THE MOVING CAR? IF YOU SAY YES, EXPLAIN WHY AND STATE ITS DIRECTION IF YOU SAY NO, EXPLAIN WHY NOT IN EITHER CASE, REFER TO YOUR FORCE DIAGRAM TO SUPPORT YOUR ANSWER [2] (D) THE TRACK IS BANKED AT AN ANGLE OF 17" AND THE CIRCULAR PATH OF THE CAR HAS RADIUS M CALCULATE THE SPEED AT WHICH THE CAR MUST TRAVEL IN ORDER THAT THERE BE NO SIDEWAYS FRICTIONAL FORCE BETWEEN THE TYRES AND THE TRACK SHOW ALL WORKING [4] COLLEGE PHYSICS FOR COLLEGE STUDENTS MULTIPLE CHOICE ITEMS 1 TWO SATELLITES OF EQUAL MASS, S1 AND S2, ORBIT THE EARTH S1 IS ORBITING AT A DISTANCE R FROM THE EARTH’S CENTRE AT SPEED V S2 ORBITS AT A DISTANCE 2R FROM THE EARTH’S CENTRE AT SPEED THE RATIO OF THE CENTRIPETAL FORCE ON S1 TO THE CENTRIPETAL FORCE ON S2 IS A B C 4 D 8 2 A POINT MASS IS MOVING IN A HORIZONTAL CIRCLE WITH A VELOCITY OF CONSTANT MAGNITUDE V AT ONE PARTICULAR TIME, THE MASS IS AT P A SHORT TIME LATER, THE MASS IS AT Q, AS SHOWN BELOW WHICH VECTOR DIAGRAM CORRECTLY SHOWS THE CHANGE IN VELOCITY ?V OF THE MASS DURING THIS TIME? P Q V V A C B D ? ? ? ? V V V V V V V V V V V V COLLEGE PHYSICS FOR COLLEGE STUDENTS 3 A PARTICLE TRAVELS AT A CONSTANT SPEED AROUND A CIRCLE OF RADIUS R WITH CENTRIPETAL ACCELERATION A WHAT IS THE TIME TAKEN FOR TEN COMPLETE ROTATIONS? A ?? 5 ? ?? ?? B ?? 5 ? ?? ?? C 20??? ?? ?? D 20??? ?? ?? 4 POINTS P AND Q ARE AT DISTANCES R AND 2R RESPECTIVELY FROM THE CENTRE X OF A DISC, AS SHOWN BELOW THE DISC IS ROTATING ABOUT AN AXIS THROUGH X, NORMAL TO THE PLANE OF THE DISC POINT P HAS LINEAR SPEED V AND CENTRIPETAL ACCELERATION A WHICH ONE OF THE FOLLOWING IS CORRECT FOR POINT Q? LINEAR SPEED CENTRIPETAL ACCELERATION A V A B V 2A C 2V 2A D 2V 4A 5 THE CENTRIPETAL FORCE F ACTING ON A PARTICLE OF MASS M THAT IS TRAVELLING WITH LINEAR SPEED V ALONG THE ARC OF A CIRCLE OF RADIUS R IS GIVEN BY A F = B F = MV2R C F = MR2V D F = Q P 2R R X MR V R MV COLLEGE PHYSICS FOR COLLEGE STUDENTS 6 FOR A PARTICLE MOVING IN A CIRCLE WITH UNIFORM SPEED, WHICH ONE OF THE FOLLOWING STATEMENTS IS CORRECT? A THE KINETIC ENERGY OF THE PARTICLE IS CONSTANT B THE FORCE ON THE PARTICLE IS IN THE SAME DIRECTION AS THE DIRECTION OF MOTION OF THE PARTICLE C THE MOMENTUM OF THE PARTICLE IS CONSTANT D THE DISPLACEMENT OF THE PARTICLE IS IN THE DIRECTION OF THE FORCE 7 A BRICK IS PLACED ON THE SURFACE OF A FLAT HORIZONTAL DISC AS SHOWN IN THE DIAGRAM BELOW THE DISC IS ROTATING AT CONSTANT SPEED ABOUT A VERTICAL AXIS THROUGH ITS CENTRE THE BRICK DOES NOT MOVE RELATIVE TO THE DISC WHICH OF THE DIAGRAMS BELOW CORRECTLY REPRESENTS THE HORIZONTAL FORCE OR FORCES ACTING ON THE BRICK? 8 THE CENTRIPETAL FORCE THAT CAUSES A CAR TO GO ROUND A BEND IN THE ROAD IS PROVIDED BY A THE FORCE PRODUCED BY THE CAR ENGINE ACTING ON THE WHEELS B THE FRICTION BETWEEN THE TYRES AND THE ROAD C THE WEIGHT OF THE CAR D THE FORCE EXERTED BY THE DRIVER ON THE STEERING WHEEL COLLEGE PHYSICS FOR COLLEGE STUDENTS 10 A PARTICLE P IS MOVING IN A CIRCLE WITH UNIFORM SPEED WHICH ONE OF THE FOLLOWING DIAGRAMS CORRECTLY SHOWS THE DIRECTION OF THE ACCELERATION A AND VELOCITY V OF THE PARTICLE AT ONE INSTANT OF TIME? 11 A MODEL CAR MOVES IN A CIRCULAR PATH OF RADIUS 0 8 M AT AN ANGULAR SPEED OF ? 6RAD S–1 WHAT IS ITS DISPLACEMENT FROM POINT P, 6 S AFTER PASSING P? A ZERO B 1 6 M C 0 4 ?M D 1 6 ?M 12 A MASS ON THE END OF A STRING IS WHIRLED ROUND IN A HORIZONTAL CIRCLE AT INCREASING SPEED UNTIL THE STRING BREAKS THE SUBSEQUENT PATH TAKEN BY THE MASS IS A A STRAIGHT LINE ALONG A RADIUS OF THE CIRCLE B A HORIZONTAL CIRCLE C A PARABOLA IN A HORIZONTAL PLANE D A PARABOLA IN A VERTICAL PLANE A B C D A A A A V V V V P P P P COLLEGE PHYSICS FOR COLLEGE STUDENTS MARKSCHEME 1 M1: AT THE SAME POSITION M2: ROTATED BY 144 DEGREES 2 28 S 3 13 3MG 14 3 2 M ; 1 8 N 15 ? 40M/S 16 ?= 2?? ; = 11?? (B) (I) TLOWEST = FC +MG = 11 + 9700 = 9711N ; UPWARD FORCE (II) THIGHEST = FC - MG = 9700 -11 = 9689N ; UPWARD FORCE (C) (I) (II) THE NORMAL FORCE IS HIGHEST AT C SINCE MG IS IN OPPOSITE DIRECTION TO CENTRIPETAL FORCE AND IT IS LOWEST AT POINT C SINCE MG IS IN THE DIRECTION OF CENTRIPERTAL FORCE COLLEGE PHYSICS FOR COLLEGE STUDENTS 18 (A) DIRECTION CHANGING; VELOCITY CHANGING SO ACCELERATING; (B) WEIGHT/GRAVITATIONAL FORCE/MG/W/FW/FG AND REACTION/NORMAL REACTION/PERPENDICULAR CONTACT FORCE/N/R/FN/FR BOTH LABELLED; (DO NOT ALLOW “GRAVITY” FOR “WEIGHT” ) WEIGHT BETWEEN WHEELS (IN BOX) FROM CENTRE OF MASS AND REACTIONS AT BOTH WHEELS / SINGLE REACTION ACTING ALONG SAME LINE OF ACTION AS THE WEIGHT; (C) G = V2/R ; =(50X9 8)-1 ; 22MS-1; ALLOW [3] FOR A BALD CORRECT ANSWER [7] 19 (A) LOOK FOR AN ANSWER ON THE FOLLOWING LINES THE DIRECTION OF THE CAR IS CHANGING; HENCE THE VELOCITY OF THE CAR IS CHANGING; OR SINCE THE DIRECTION OF THE CAR IS CHANGING; A FORCE MUST BE ACTING ON IT, HENCE IT IS ACCELERATING; 2 MAX (B) (I) ARROW POINTING VERTICALLY DOWNWARDS; (II) WEIGHT; DO NOT PENALIZE THE CANDIDATE IF THEY STATE “GRAVITY” NORMAL REACTION; DO NOT PENALIZE THE CANDIDATE IF THEY STATE “PUSH OF THE TRACK ON THE MARBLE” (III) LOSS IN PE = 0 05 × 10 × (0 8 – 0 35); = GAIN IN KE = MV2; TO GIVE V = 3 0 M S–1; OR USE OF V = TO GIVE V = 4 0 M S–1 AT POINT B; AND THEN USE OF V2 – U2 = 2GH WITH V = 4 0 M S–1 AND H = 0 35 M; TO GET U = 3 0 M S–1; 3 MAX DO NOT PENALIZE THE CANDIDATE IF G = 9 8 M S–2 IS USED (IV) RECOGNIZE THAT RESULTANT FORCE = ; = = 2 6 N; N = – MG; = 2 6 – 0 5 = 2 1 N; [12] R MV ( ? R MV COLLEGE PHYSICS FOR COLLEGE STUDENTS 20 (A) YES IT HAS ACCELERATION BECAUSE ITS VELOCITY DIRECTION IS CHANGING [1] – TOWARD THE CENTRE OF THE TRACK (B)] (AWARD [1] FOR WEIGHT DOWNWARD, [1] FOR NORMAL FORCE PERPENDICULAR TO TRACK SUBTRACT MARKS FOR NON-EXISTENT OR WRONG FORCES ) (C) YES, THERE IS A RESULTANT FORCE: THE CAR IS NOT IN EQUILIBRIUM, BUT ACCELERATING TOWARD THE CENTRE OF THE CIRCULAR TRACK [1] THE RESULTANT FORCE ON IT IS TOWARD THE CENTRE, AND IN THE FORCE DIAGRAM IT WOULD BE THE RESULTANT OF THE TWO FORCES SHOWN [10] KEY FOR MULTIPLE CHOICE ITEMS: 1 C 2 C 3 D 4 C 5 D 6 A 7 D 8 B 9 D 10 C 11 B 12 D COLLEGE PHYSICS FOR COLLEGE STUDENTS UNIVERSAL GRAVITATION PROBLEMS 1 THIS QUESTION IS ABOUT GRAVITATION AND ORBITAL MOTION (A) DEFINE GRAVITATIONAL FIELD STRENGTH AT A POINT IN A GRAVITATIONAL FIELD (2) THE DIAGRAM BELOW SHOWS THREE POINTS ABOVE A PLANET THE ARROW REPRESENTS THE GRAVITATIONAL FIELD STRENGTH AT POINT A (B) DRAW ARROWS TO REPRESENT THE GRAVITATIONAL FIELD STRENGTH AT POINT B AND POINT C (2) (TOTAL 4 MARKS) 2 THIS QUESTION IS ABOUT GRAVITATIONAL FIELDS (A) DEFINE GRAVITATIONAL FIELD STRENGTH (2) THE GRAVITATIONAL FIELD STRENGTH AT THE SURFACE OF JUPITER IS 25 N KG–1 AND THE RADIUS OF JUPITER IS 7 1 × 107 M (B) (I) DERIVE AN EXPRESSION FOR THE GRAVITATIONAL FIELD STRENGTH AT THE SURFACE OF A PLANET IN TERMS OF ITS MASS M, ITS RADIUS R AND THE GRAVITATIONAL CONSTANT G (2) (II) USE YOUR EXPRESSION IN (B)(I) ABOVE TO ESTIMATE THE MASS OF JUPITER (2) (TOTAL 6 MARKS) A PLANET B C COLLEGE PHYSICS FOR COLLEGE STUDENTS 3 THIS QUESTION IS ABOUT A SPACECRAFT A SPACECRAFT ABOVE EARTH’S ATMOSPHERE IS MOVING AWAY FROM THE EARTH THE DIAGRAM BELOW SHOWS TWO POSITIONS OF THE SPACECRAFT POSITION A AND POSITION B ARE WELL ABOVE EARTH’S ATMOSPHERE AT POSITION A, THE ROCKET ENGINE IS SWITCHED OFF AND THE SPACECRAFT BEGINS COASTING FREELY AT POSITION A, THE SPEED OF THE SPACECRAFT IS 5 37 X 103 M S–1 AND AT POSITION B, 5 10 X 103 M S–1 THE TIME TO TRAVEL FROM POSITION A TO POSITION B IS 6 00 X 102 S (A) (I) EXPLAIN WHY THE SPEED IS CHANGING BETWEEN POSITIONS A AND B (1) (II) CALCULATE THE AVERAGE ACCELERATION OF THE SPACECRAFT BETWEEN POSITIONS A AND B (2) (III) ESTIMATE THE AVERAGE GRAVITATIONAL FIELD STRENGTH BETWEEN POSITIONS A AND B EXPLAIN YOUR WORKING (3) (B) THE DIAGRAM BELOW SHOWS THE VARIATION WITH DISTANCE FROM EARTH OF THE KINETIC ENERGY EK OF THE SPACECRAFT THE RADIUS OF EARTH IS R ON THE DIAGRAM ABOVE, DRAW THE VARIATION WITH DISTANCE FROM THE SURFACE OF EARTH OF THE GRAVITATIONAL POTENTIAL ENERGY EP OF THE SPACECRAFT (2) (TOTAL 8 MARKS) EARTH A B ENERGY EK R 0 DISTANCE COLLEGE PHYSICS FOR COLLEGE STUDENTS 4 THIS QUESTION IS ABOUT GRAVITATION (A) (I) DEFINE GRAVITATIONAL POTENTIAL AT A POINT IN A GRAVITATIONAL FIELD (2) (II) EXPLAIN WHY VALUES OF GRAVITATIONAL POTENTIAL HAVE NEGATIVE VALUES (2) THE EARTH AND THE MOON MAY BE CONSIDERED TO BE TWO ISOLATED POINT MASSES THE MASSES OF THE EARTH AND THE MOON ARE 5 98 X 1024KG AND 7 35 X 1022 KG RESPECTIVELY AND THEIR SEPARATION IS 3 84 X 108 M, AS SHOWN BELOW THE DIAGRAM IS NOT TO SCALE (B) (I) DEDUCE THAT, AT POINT P, 3 46 X 108M FROM EARTH, THE GRAVITATIONAL FIELD STRENGTH IS APPROXIMATELY ZERO (3) (II) THE GRAVITATIONAL POTENTIAL AT P IS ?1 28 X 106 J KG–1 CALCULATE THE MINIMUM SPEED OF A SPACE PROBE AT P SO THAT IT CAN ESCAPE FROM THE ATTRACTION OF THE EARTH AND THE MOON (3) (TOTAL 10 MARKS) EARTH MOON MASS = 5 98 MASS = 7 35 P 3 84 5 THIS QUESTION IS ABOUT GRAVITATIONAL POTENTIAL (A) DEFINE GRAVITATIONAL POTENTIAL AT A POINT IN A GRAVITATIONAL FIELD (3) (B) A PLANET HAS MASS M AND RADIUS R0 THE MAGNITUDE G0 OF THE GRAVITATIONAL FIELD STRENGTH AT THE SURFACE OF A PLANET IS WHERE G IS THE GRAVITATIONAL CONSTANT USE THIS EXPRESSION TO DEDUCE THAT THE GRAVITATIONAL POTENTIAL V0 AT THE SURFACE OF THE PLANET IS GIVEN BY V0 = – G0R0 (2) (C) THE GRAPH BELOW SHOWS THE VARIATION WITH DISTANCE R FROM THE CENTRE OF THE PLANET OF THE GRAVITATIONAL POTENTIAL V THE RADIUS R0 OF THE PLANET = 5 0 X 106 M VALUES OF V ARE NOT SHOWN FOR R = R0 USE THE GRAPH TO DETERMINE THE MAGNITUDE OF THE GRAVITATIONAL FIELD STRENGTH AT THE SURFACE OF THE PLANET (3) R / 10 M V / 10 JKG (D) A SATELLITE OF MASS 3 2 X 103 KG IS LAUNCHED FROM THE SURFACE OF THE PLANET USE THE GRAPH TO ESTIMATE THE MINIMUM LAUNCH SPEED THAT THE SATELLITE MUST HAVE IN ORDER TO REACH A HEIGHT OF 2 0 X 107 M ABOVE THE SURFACE OF THE PLANET (YOU MAY ASSUME THAT IT REACHES ITS MAXIMUM SPEED IMMEDIATELY AFTER LAUNCH ) (4) (TOTAL 12 MARKS) 6 THIS QUESTION IS ABOUT THE GRAVITATIONAL FIELD OF MARS (A) DEFINE GRAVITATIONAL POTENTIAL ENERGY OF A MASS AT A POINT (1) (B) THE GRAPH SHOWS THE VARIATION WITH DISTANCE R FROM THE CENTRE OF MARS OF THE GRAVITATIONAL POTENTIAL V R IS THE RADIUS OF MARS WHICH IS 3 3 MM (VALUES OF V FOR R < R ARE NOT SHOWN ) A ROCKET OF MASS 1 2 × 104 KG LIFTS OFF FROM THE SURFACE OF MARS USE THE GRAPH TO (I) CALCULATE THE CHANGE IN GRAVITATIONAL POTENTIAL ENERGY OF THE ROCKET AT A DISTANCE 4R FROM THE CENTRE OF MARS (3) (II) SHOW THAT THE MAGNITUDE OF THE GRAVITATIONAL FIELD STRENGTH AT A DISTANCE 4R FROM THE CENTRE OF MARS IS 0 23 N KG–1 (2) COLLEGE PHYSICS FOR COLLEGE STUDENTS (C) USE THE ANSWER TO (B)(II) TO SHOW THAT THE MAGNITUDE OF THE GRAVITATIONAL FIELD STRENGTH AT THE SURFACE OF MARS IS 3 7 N KG–1 (2) (D) THE GRAVITATIONAL POTENTIAL AT THE SURFACE OF EARTH IS –63 MJ KG–1 WITHOUT ANY FURTHER CALCULATION, COMPARE THE ESCAPE SPEED REQUIRED TO LEAVE THE SURFACE OF EARTH WITH THAT OF THE ESCAPE SPEED REQUIRED TO LEAVE THE SURFACE OF MARS (2) (TOTAL 10 MARKS) 7 THIS QUESTION IS ABOUT GRAVITATION A SPACE PROBE IS LAUNCHED FROM THE EQUATOR IN THE DIRECTION OF THE NORTH POLE OF THE EARTH DURING THE LAUNCH, THE ENERGY E GIVEN TO THE SPACE PROBE OF MASS M IS E = WHERE G IS THE GRAVITATIONAL CONSTANT AND M AND RE ARE, RESPECTIVELY, THE MASS AND RADIUS OF THE EARTH WORK DONE IN OVERCOMING FRICTIONAL FORCES IS NOT TO BE CONSIDERED (A) (I) EXPLAIN WHAT IS MEANT BY ESCAPE SPEED (2) (II) DEDUCE THAT THE SPACE PROBE WILL NOT BE ABLE TO TRAVEL INTO DEEP SPACE (3) E 4R GMM COLLEGE PHYSICS FOR COLLEGE STUDENTS THE SPACE PROBE IS LAUNCHED INTO A CIRCULAR POLAR ORBIT OF RADIUS R (B) DERIVE EXPRESSIONS, IN TERMS OF G, M, RE, M AND R, FOR (I) THE CHANGE IN GRAVITATIONAL POTENTIAL ENERGY OF THE SPACE PROBE AS A RESULT OF TRAVELLING FROM THE EARTH’S SURFACE TO ITS ORBIT (1) (II) THE KINETIC ENERGY OF THE SPACE PROBE WHEN IN ITS ORBIT (2) (C) USING YOUR ANSWERS IN (B) AND THE TOTAL ENERGY SUPPLIED TO THE SPACE PROBE AS GIVEN IN (A), DETERMINE THE HEIGHT OF THE ORBIT ABOVE THE EARTH’S SURFACE (4) A SPACE PROBE IN A LOW ORBIT ROUND THE EARTH WILL EXPERIENCE FRICTION DUE TO THE EARTH’S ATMOSPHERE (D) (I) DESCRIBE HOW FRICTION WITH THE AIR REDUCES THE ENERGY OF THE SPACE PROBE (2) (II) SUGGEST WHY THE RATE OF LOSS OF ENERGY OF THE SPACE PROBE DEPENDS ON THE DENSITY OF THE AIR AND ALSO THE SPEED OF THE SPACE PROBE (2) (III) STATE WHAT WILL HAPPEN TO THE HEIGHT OF THE SPACE PROBE ABOVE THE EARTH’S SURFACE AND TO ITS SPEED AS AIR RESISTANCE GRADUALLY REDUCES THE TOTAL ENERGY OF THE SPACE PROBE (2) (TOTAL 18 MARKS) COLLEGE PHYSICS FOR COLLEGE STUDENTS 8 THIS QUESTION IS ABOUT ESCAPE SPEED AND KEPLER’S THIRD LAW JUPITER AND EARTH ARE TWO PLANETS THAT ORBIT THE SUN THE EARTH HAS MASS ME AND DIAMETER DE THE ESCAPE SPEED FROM EARTH IS 11 2 KM S–1 DATA FOR JUPITER ARE GIVEN BELOW MASS: 1 90 × 1027 KG (318 ME) MEAN DIAMETER: 1 38 × 105 KM (10 8 DE) (A) (I) STATE WHAT IS MEANT BY ESCAPE SPEED (1) (II) ESCAPE SPEED V IS GIVEN BY THE EXPRESSION V = DETERMINE THE ESCAPE SPEED FROM JUPITER (2) (B) (I) STATE KEPLER’S THIRD LAW (1) (II) IN 1610, THE MOON GANYMEDE WAS DISCOVERED ORBITING JUPITER ITS ORBIT WAS FOUND TO HAVE A RADIUS OF 15 0 R AND PERIOD 7 15 DAYS, WHERE R IS THE RADIUS OF JUPITER ANOTHER MOON OF JUPITER, LYSITHEA, WAS DISCOVERED IN 1938 AND ITS ORBIT WAS FOUND TO HAVE A RADIUS OF 164 R AND A PERIOD OF 260 DAYS SHOW THAT THESE DATA ARE CONSISTENT WITH KEPLER’S THIRD LAW (2) (TOTAL 6 MARKS) ? ? ? ? ? ? R GM 9 MOTION OF A SATELLITE (A) DEFINE GRAVITATIONAL POTENTIAL (2) (B) A SATELLITE OF MASS M IS IN A CIRCULAR ORBIT AROUND THE EARTH AT HEIGHT R FROM THE EARTH’S SURFACE THE MASS OF THE EARTH MAY BE CONSIDERED TO BE A POINT MASS CONCENTRATED AT THE EARTH’S CENTRE THE EARTH HAS MASS M AND RADIUS R (I) DEDUCE THAT THE KINETIC ENERGY EK OF THE SATELLITE WHEN IN ORBIT OF HEIGHT R IS (3) (II) THE KINETIC ENERGY OF THE SATELLITE IN THIS ORBIT IS 1 5 X 1010 J CALCULATE THE TOTAL ENERGY OF THE SATELLITE (3) ORBIT SATELLITE MASS M R R EARTH MASS M K R GMM E = COLLEGE PHYSICS FOR COLLEGE STUDENTS (III) EXPLAIN HOW YOUR ANSWER TO (B)(II) INDICATES THAT THE SATELLITE WILL NOT ESCAPE THE EARTH’S GRAVITATIONAL FIELD AND STATE THE MINIMUM AMOUNT OF ENERGY THAT MUST BE PROVIDED TO THIS SATELLITE SO THAT IT DOES ESCAPE (3) (TOTAL 11 MARKS) 10 THIS QUESTION IS ABOUT A SATELLITE ORBITING THE EARTH A SATELLITE S IS IN ORBIT ROUND THE EARTH, A DISTANCE R = 4 2 × 107 M FROM THE CENTRE OF THE EARTH (A) ON THE DIAGRAM ABOVE, FOR THE SATELLITE IN THE POSITION SHOWN, DRAW ARROW(S) TO REPRESENT THE FORCE(S) ACTING ON THE SATELLITE (1) (B) DEDUCE THAT THE VELOCITY V OF THE SATELLITE IS GIVEN BY THE EXPRESSION V2 = WHERE M IS THE MASS OF THE EARTH (1) R = 4 2 × 10 M EARTH S R GM COLLEGE PHYSICS FOR COLLEGE STUDENTS (C) HENCE DEDUCE THAT THE PERIOD OF ORBIT T OF THE SATELLITE IS GIVEN BY THE FOLLOWING EXPRESSION (3) (D) USE THE FOLLOWING INFORMATION TO DETERMINE THAT THE ORBITAL PERIOD OF THE SATELLITE IS ABOUT 24 HOURS ACCELERATION DUE TO GRAVITY AT THE SURFACE OF THE EARTH G = = 10 MS?2, WHERE M IS THE MASS OF THE EARTH AND RE IS THE RADIUS OF THE EARTH = 6 4 × 106 M (2) (E) THE SATELLITE IS MOVED INTO AN ORBIT THAT IS CLOSER TO THE EARTH STATE WHAT HAPPENS TO ITS (I) POTENTIAL ENERGY (1) (II) KINETIC ENERGY (1) (TOTAL 9 MARKS) GM R T 4? = E R GM COLLEGE PHYSICS FOR COLLEGE STUDENTS (I) DEDUCE THAT THE GRAVITATIONAL FIELD STRENGTH OF JUPITER AT THE SURFACE OF GANYMEDE IS APPROXIMATELY 0 1 N KG?1 (2) (II) ESTIMATE THE MASS OF JUPITER (3) (TOTAL 12 MARKS) GM K ? = COLLEGE PHYSICS FOR COLLEGE STUDENTS R R R GM T + ? = COLLEGE PHYSICS FOR COLLEGE STUDENTS MARKSCHEME 1 (A) G = F IS THE GRAVITATIONAL FORCE; EXERTED ON / EXPERIENCED BY A SMALL / POINT / INFINITESIMAL MASS M; (B) AWARD [1] FOR EACH CORRECT ARROW THE ONE AT B POINTS IN THE SAME DIRECTION AS THAT AT A AND IS SHORTER THE ONE AT C HAS THE SAME LENGTH AS THAT AT A AND POINTS TOWARD THE CENTRE OF THE PLANET [4] 2 (A) THE GRAVITATIONAL FORCE EXERTED PER UNIT MASS; ON A POINT / SMALL MASS; (B) (I) USE OF G = AND F = G ; COMBINE TO GET G = G ; G ; SUBSTITUTE TO GET M = 1 9 × 1027 KG; [6] 3 (A) (I) (DECELERATION DUE TO) GRAVITATIONAL PULL OF EARTH; (II) A = - 0 45MS-2; (III) ECF FROM (II): E = A; E = - 0 45NK G-1; ACCEPT MS-2 AS CORRECT UNITS (B) GENERAL SHAPE (1 / R); CORRECT QUADRANT; [8] M F C A B PLANET M F R MM R M ; 5100? = ? ? = T V A ; M F E = COLLEGE PHYSICS FOR COLLEGE STUDENTS 4 (A) (I) WORK DONE PER UNIT MASS; IN MOVING (SMALL MASS) FROM INFINITY TO THAT POINT; (II) GRAVITATIONAL FORCES ARE ALWAYS ATTRACTIVE; WORK GOT OUT WHEN MOVING FROM INFINITY; WORK DONE AGAINST FIELD IS NEGATIVE; (B) (I) (RATIO IDEA IS ESSENTIAL) SOME COMMENT EG SHOULD BE SAME SO QED; OR: (RATIO IDEA IS ESSENTIAL) SOME COMMENT EG ABOUT SAME SO QED; OR: (RATIO IDEA IS ESSENTIAL) R = 3 45 X 108 M; (II) INDICATES IN SOME WAY THAT ?GPE IS CHANGE IN KINETIC ENERGY; GRAVITATIONAL POTENTIAL = V = 1 6 X 103 MS-1; [10] 5 (A) THE WORK DONE PER UNIT MASS; IN BRINGING A SMALL / POINT MASS; FROM INFINITY TO THE POINT (IN THE GRAVITATIONAL FIELD); (B) GM = G0 R02 TO GIVE V0 = - G0R0; = 7 8(±0 2) NKG-1; ; 4 M E = M M ( ) ( ) ; 9 R M ( ) ( ) ; EARTH G G F ? = ? ? ? = ( ) ( ) ; MOON G G F ? = ? ? ? = ( ) ; E R GM R GM ? ? = ( ) ; R R ? ? = ; 1 V ; R M G V ? = R M G V ? = COLLEGE PHYSICS FOR COLLEGE STUDENTS (D) 2 0 X 107 M ABOVE SURFACE IS 2 5 X 107 M FROM CENTRE; ?V BETWEEN SURFACE AND 2 5 X 107M = (- 0 8 – (- 3 9)) X = - 3 1 X 107 J KG-1; ??????= ????? ? ?????= AWARD [3 MAX] IF THE CANDIDATE FORGETS THAT THE DISTANCES ARE FROM THE CENTRE, THE CANDIDATE MUST SHOW ?V [12] 6 (A) WORK DONE IN MOVING MASS FROM INFINITY TO A POINT; (B) (I) READ OFFS –12 6 AND –3 2; GAIN IN GPE 1 2 × 104 × [12 6 – 3 2] OR GAIN IN G POTENTIAL [12 6 × 106 – 3 2 × 106] = 1 13 ± 0 05 × 105 MJ OR = 1 13 + 0 05 × 1011 J; (II) USE OF GRADIENT OF GRAPH TO DETERMINE G; VALUES SUBSTITUTED FROM DRAWN GRADIENT ; = 0 23 N KG–1 (ALLOW ANSWERS IN THE RANGE OF 0 20 TO 0 26 N KG–1) AWARD [0] FOR SOLUTIONS FROM (C) G AT SURFACE = 42 G AT 4R; AND = 16 1 OR 16 × 0 23 = 3 7 N KG–1; (D) ESCAPE SPEED FOR EARTH > ESCAPE SPEED FOR MARS; POTENTIAL LESS/MORE NEGATIVE AT EARTH; [10] 7 (A) (I) SPEED OF OBJECT AT EARTH’S SURFACE; SO THAT IT WILL ESCAPE FROM THE GRAVITATIONAL FIELD / TRAVEL TO INFINITY; (II) GRAVITATIONAL POTENTIAL ENERGY AT EARTH’S SURFACE = (–) ; THIS MUST BE PROVIDED FOR PROBE TO ESCAPE; ENERGY IS LESS THAN THIS HENCE NOT ESCAPE; (B) (I) CHANGE = GMM ; ACCEPT GMM (II) IN ORBIT, ; ; (C) IDEA OF EQUATING ENERGIES; ?? ? ? ?? ? ? ? ? ? TYPICALLY R V E R GMM ?? ? ? ?? ? ? ?R R E ?? ? ? ?? ? ? ? E R R R GMM R MV = R GMM MV = COLLEGE PHYSICS FOR COLLEGE STUDENTS R = 2RE; HEIGHT ABOVE SURFACE = RE; 4 MAX (D) (I) PROBE COLLIDES WITH AIR MOLECULES; GIVING THEM KINETIC ENERGY AND SO LOSING ENERGY ITSELF; ACCEPT ANSWERS IN TERMS OF FRICTIONAL FORCES (II) GREATER DENSITY, MORE MOLECULES OF AIR WITH WHICH TO COLLIDE; HIGHER SPEED, HIGHER REBOUND SPEED FOR AIR MOLECULES; ACCEPT ANSWERS IN TERMS OF MAGNITUDE OF FRICTIONAL FORCE (III) HEIGHT BECOMES LESS; AND SPEED INCREASES; [18] 8 (A) (I) SPEED WITH WHICH OBJECT MUST LEAVE (SURFACE OF PLANET) TO (COMPLETELY) ESCAPE ITS GRAVITATIONAL FIELD; (II) V = 11 2 × OR FULL SUBSTITUTION; = 60 8 KM S–1; (B) (I) T2 ? R3 WITH SYMBOLS EXPLAINED / IN WORDS; (II) FOR GANYMEDE, = 0 0151 FOR LYSTHEA, = 0 0153; DO NOT AWARD THE RATIOS IF ONLY 2 SIGNIFICANT FIGURE ARE GIVEN THIS IS NOT THE SIGNIFICANT FIGURE MARK SUITABLE COMMENT BASED ON CANDIDATE’S CALCULATIONS; 2 MAX [6] 9 MOTION OF A SATELLITE (A) THE WORK DONE PER UNIT MASS; IN BRINGING A SMALL / POINT MASS FROM INFINITY TO A POINT (IN THE GRAVITATIONAL FIELD); RATIO IDEA ESSENTIAL FOR FIRST MARK (B) (I) EQUATING GRAVITATIONAL FORCE TO CENTRIPETAL TO GET THE SPEED AND HENCE EK = R GMM R GMM R GMM R GMM ? + = E E E = 8 R MV R GMM = ; R GM V = ; R GMM R GMM R GMM = = COLLEGE PHYSICS FOR COLLEGE STUDENTS (II) POTENTIAL ENERGY IN ORBIT IS ; TOTAL ENERGY IS THEN IE E = -1 5 X 1010J; AWARD [1 MAX] FOR BARE ANSWER WITHOUT EXPLANATION (III) AT INFINITY THE POTENTIAL ENERGY IS ZERO AND HENCE IF SATELLITE GETS THERE ITS TOTAL ENERGY WILL BE E = 0; BUT THE SATELLITE HAS NEGATIVE ENERGY AND HENCE IT CANNOT ESCAPE TO INFINITY / THE SATELLITE IS BOUND TO THE EARTH FOREVER; [11] 10 (A) F – TOWARDS CENTRE OF EARTH; 1 MAX AWARD [0] IF ANY OTHER FORCES ARE DRAWN (B) SO V2 = ; 1 MAX (C) V = ; SO V2 = ; SO T2 = ; 3 MAX (D) T2 = ; TO GIVE T = 85000 S = 24 HOURS; 2 MAX (E) (I) DECREASES; 1 MAX (II) INCREASES; 1 MAX [9] R GMM R GMM E P ? = ? = ; R GMM R GMM R GMM E ? = ? = ( )( ) ( ) (B) (II) ANSWER TO CANDIDATES WITH THE MUST AGREE THE VALUE II B FROM ECF OR E ? ? ? ? = ;J IS SUPPLIED BE MUST T ENERGY THA MINIMUM THE R V R GM 2 = R GM T R ? R GM T R = GM R E R G R R = 4 2 × 10 M EARTH S F COLLEGE PHYSICS FOR COLLEGE STUDENTS 11 KEPLER’S THIRD LAW (A) (I) THE ORBITS ARE ELLIPTICAL / NOT CIRCULAR; DO NOT ACCEPT “BECAUSE R CHANGES” (II) GRAVITY / GRAVITATIONAL; (III) SAME SYMBOLS AS IN QUESTION MUST BE USED TO RECEIVE THE MARK (IV) LET V = THE SPEED OF PLANET, THE ACCELERATION IS THEN = FROM NEWTON 2, THEREFORE, = = THEREFORE, SO (II) THEREFORE, 2 0 X 1027 KG; OR = = 2 0 = 1027 KG; [12] ; ? ? ? = = RK VE THIS MA D TO RECEI IS REQUIRE WTON ENCE TO NE SOME REFER R MV R M M G F S ; T R V ? = R M M G S ; T MR ? ; R GM T S ? = S GM K T R R A ? = ? = ; T R R V ? = ( ) ; R GM T S ? = ; GT R M JUP ? = ( ) ( ) ; G R M = ( ) ? ? ? ? COLLEGE PHYSICS FOR COLLEGE STUDENTS 12 (A) GRAVITATION / GRAVITY; (B) GRAVITATIONAL FORCE = CENTRIPETAL FORCE = GRAVITATIONAL FORCE PROVIDES CENTRIPETAL FORCE (C) FROM FORMULA, IS A CONSTANT; (SO IF R1 IS SMALLER) THEN M2 IS SMALLER / M1 IS LARGER; DO NOT AWARD SECOND MARK IF NO REASONING GIVEN OR ARGUMENT IS FALLACIOUS [6] ( ) ; R R M GM + ; T R M ? ? ( ) ; T R M R R M GM ? ? = + ( ) GM R R R T ? ? + = M R COLLEGE PHYSICS FOR COLLEGE STUDENTS