Revision Guide fo Chapte 11 Contents Revision Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Wok... 5 Gavitational field... 5 Potential enegy... 7 Kinetic enegy... 8 Pojectile... 9 Gavitational potential... 10 Cicula motion... 11 Keple's laws... 1 Satellite motion... 14 Summay Diagams Consevation of momentum... 16 Two caft collide... 17 Moe collisions... 19 Momentum, invaiance, symmety... 1 Jets and ockets... Gaphs showing g against h... 3 Field and potential... 4 Centipetal acceleation... 6 Geomety ules the Univese... 7 A geostationay satellite... 8 Relationship between g and V g... 9 Speeds and acceleations in the Sola System... 30 Acceleation of the Moon... 31 Apollo etuns fom the Moon... 3 Advancing Physics A 1
Revision Checklist Back to list of Contents I can show my undestanding of effects, ideas and elationships by descibing and explaining cases involving: momentum as the poduct of mass velocity foce defined as ate of change of momentum consevation of momentum when objects inteact; Newton's thid law as a consequence Revision Notes: momentum, Newton's laws of motion Summay Diagams: Consevation of momentum, Two caft collide, Moe collisions, Momentum, invaiance, symmety, Jets and ockets wok done (as foce distance moved in the diection of the foce: including cases whee the foce does not act in the diection of the esulting motion) changes of gavitational potential enegy to kinetic enegy and vice vesa when objects move in a gavitational field motion in a unifom gavitational field Revision Notes: wok, gavitational field, potential enegy, kinetic enegy, pojectile Summay Diagams: Gaph showing g against h the gavitational field and gavitational potential due to a point mass Revision Notes: gavitational field, gavitational potential Summay Diagams: Field and potential motion in a hoizontal cicle and in a cicula gavitational obit about a cental mass Revision Notes: cicula motion, Keple's laws Summay Diagams: Centipetal acceleation, Geomety ules the Univese, A geostationay satellite I can use the following wods and phases accuately when descibing effects and obsevations: foce, momentum Revision Notes: momentum kinetic enegy, potential enegy Revision Notes: kinetic enegy, potential enegy gavitational field, gavitational potential, equipotential suface Revision Notes: gavitational field, gavitational potential Summay Diagams: Relationship between g and V g Advancing Physics A
I can sketch, plot and intepet: gaphs showing the vaiation of a gavitational field with distance, and know that the aea unde the gaph shows the change in gavitational potential Revision Notes: gavitational field Summay Diagams: Field and potential, Gaph showing g against h gaphs showing the vaiation of gavitational potential with distance, and know that the tangent to the cuve gives the gavitational field stength Revision Notes: gavitational potential Summay Diagams: Field and potential diagams illustating gavitational fields and the coesponding equipotential sufaces Summay Diagams: Relationship between g and V g, Field and potential I can make calculations and estimates involving: kinetic enegy ½ mv, gavitational potential enegy change mgh enegy tansfes and exchanges using the idea: wok done ΔE = FΔs cosθ (no wok is done when F and s ae pependicula) Revision Notes: kinetic enegy, potential enegy, gavitational potential, pojectile momentum p =mv and F = Δ(mv) / Δt Revision Notes: momentum, Newton's laws of motion cicula and obital motion: a = v / ; F = mv / Revision Notes: satellite motion, Keple's laws Summay Diagams: Centipetal acceleation, Speeds and acceleations in the Sola System, Acceleation of the Moon, A geostationay satellite gavitational fields: fo the adial components F gav GmM F = g, m GM gav = = gavitational potential enegy gavitational potential V gav GmM E = m gav GM = Revision Notes: gavitational field, gavitational potential Summay Diagams: Field and potential, Apollo etuns fom the Moon Advancing Physics A 3
Revision Notes Back to list of Contents Momentum Momentum is mass velocity. Momentum is a vecto quantity. The SI unit of momentum is kg m s 1. Newton's second law defines the magnitude of the foce as the magnitude of the ate of change of momentum, with the foce in the diection of the change of momentum Δ( mv) F =. Δt If the mass is constant this can be expessed as 'foce = mass acceleation' because acceleation is ate of change of velocity. The change of momentum of an object acted on by a foce is: Δ ( mv) = FΔt. The poduct FΔ t is called the impulse of the foce. The thust on a ocket of the jet of gases that it ejects is equal to the ate at which the jet caies away momentum. This is given by the mass ejected pe second x the velocity of the jet. When two objects inteact, fo example in a collision, one object loses an amount of momentum and the othe object gains an equal amount. The total momentum of the two objects is the same afte the inteaction as befoe. This is the pinciple of consevation of momentum. Since the time of inteaction Δt is the same fo both objects, the foces acting on the objects ae equal and opposite. This is Newton s thid law. It is a consequence of the consevation of momentum. Newton's laws of motion Newton's fist law of motion states that an object emains at est o moves with constant velocity unless acted on by a esultant foce. Newton's fist law defines what a foce is, namely any physical effect that is capable of changing the motion of an object. If an object is at est o in unifom motion, eithe no foce acts on it o foces do act on it and the esultant foce is zeo. Newton's second law of motion states that the magnitude of the ate of change of momentum of an object is equal to the magnitude of the esultant foce on the object, with the change of momentum in the diection of the foce. That is, F = dp / dt, whee p = mv is the momentum of an object acted on by a esultant foce F. Fo an object of constant mass m, acted on by a foce F Advancing Physics A 4
dv F = m = ma dt The SI unit of foce is the newton (N). 1 N is the foce that gives a 1 kg mass an acceleation of 1 m s. Newton's thid law of motion states that when two objects inteact, thee is an equal and opposite foce (of the same kind of foce) on each object. So e.g. fo a book at est on a table, the Newton pai of foces on the book ae its weight and the gavitational foce the book exets upon the planet Eath (i.e. not the eaction foce of the table on the book, which is not a gavitational foce). Wok Wok is the change in enegy when a foce moves in the diection of that foce. The wok W done by a foce of magnitude F that moves its point of application by a distance s in the diection of the foce is given by W = F s. Wok is a pimay means of measuing amounts of enegy tansfeed fom one thing to anothe. The SI unit of enegy and of wok is the joule (J). Wok and enegy ae scala quantities. Wok is done wheneve an object moves unde the action of a foce with a component of the foce along the displacement. If thee is no movement of the object, no wok is done. Howeve, an outstetched hand holding a book does need to be supplied with enegy to keep the am muscles taut. No wok is done on the book povided it is stationay. But, enegy has to be supplied to epeatedly contact the muscle fibes so as to keep the muscles taut. No wok is done on an object by a foce when the displacement of the object is at ight angles to the diection of the foce. If an object is moved by a foce F a distance s along a line that is at angle θ to the diection of the foce, the wok done by the foce is given by W = Fs cosθ. Wok done displacement s Foce F θ Wok done = Fs cos θ Gavitational field The stength g of a gavitational field at a point is the gavitational foce pe unit mass acting on a small mass at that point. Gavitational field stength is a vecto quantity in the diection of the gavitational foce. Advancing Physics A 5
The SI unit of gavitational field stength is N kg 1 o equivalently m s. The magnitude of the gavitational foce F on a mass m at a point in a gavitational field (i.e. the objects weight) is given by F = m g, whee g is the magnitude of the gavitational field stength at that point. Close to the suface of the Eath, the gavitational field is almost unifom. The lines of foce ae paallel and at ight angles to the Eath's suface. A unifom gavitational field unifom field ove distance << adius R spheical planet of adius R adial field On a lage scale, the gavitational field is adial. Newton's law of gavitation states that the adial component of the foce of gavitational attaction F of a mass M on anothe mass m obeys an invese squae law: GMm F = whee is the distance fom the cente of M to m and the minus sign indicates that the adial component of the foce acts towads the mass M. The measued value of the Univesal Gavitational Constant G is 6.67 10 11 N m kg. The adial component of the gavitational field stength due to M is g = F / m = G M / at distance fom the cente of the mass M. Advancing Physics A 6
Vaiation of g with distance fom the cente of the Eath 10 8 6 4 0 1 3 4 5 6 distance fom cente adius of Eath Potential enegy The potential enegy of a system is the enegy associated with the position of objects elative to one anothe, fo example a mass aised above the Eath. The SI unit of potential enegy is the joule. Potential enegy can be thought of as stoed in a field, fo example a gavitational field. The potential enegy is measued by the capacity to do wok if positions of objects change. Fo example, if the potential enegy of two magnets at est epelling each othe was 100 J, then eleasing the magnets would enable them to use the potential enegy in the field to do wok. If the height of an object above the Eath changes, its potential enegy changes. Since the foce of gavity on an object of mass m due to the Eath is equal to mg, the enegy tansfeed when an object of mass m is aised a height h above the Eath = foce distance moved along the line of action of the foce = mgh, whee g is the gavitational field stength. Hence the change of potential enegy = mgh (povided the field g does not vay appeciably with height h). Advancing Physics A 7
Gain of potential enegy h mg mg ΔE p = mgh Potential enegy can be negative as well as positive. The potential enegy of a 1 kg mass and the Eath, if the mass is at the suface of the Eath, is 64 MJ. The negative sign means that the object is lowe in enegy at the suface than it is futhe away fom the Eath. In this case the 'zeo' of potential has been set at a vey geat distance ('infinity') fom the Eath. Change of potential enegy = m g h, fo a small change of height h of an object of mass m. Kinetic enegy The kinetic enegy of a moving object is the enegy it caies due to its motion. Fo an object of mass m moving at speed v, the kinetic enegy E k = ½ m v, povided its speed is much less than the speed of light. If an object of mass m initially at est is acted on by a constant foce of magnitude F fo a time t, the object acceleates to speed v whee F t = m v. Since the distance moved by the object, s = ½ v t, then the wok done on the object is equal to mv vt Fs = = t 1 mv. The wok done is equal to the gain of kinetic enegy. Hence the kinetic enegy at speed v is E k = ½ m v. Note that the elationship is an appoximation fo speeds which ae small compaed with the speed of light. At all speeds, fom the theoy of elativity, the kinetic enegy of a fee paticle may be defined as E k = E total E est, whee E est, the est enegy, is given by E est = m c, and E total, the total enegy, is given by γ m c. Hee γ is the elativistic facto γ = 1 (see chapte 1). 1 v / c Relationships E k = ½ m v Advancing Physics A 8
Pojectile A pojectile is any object in motion close to the Eath and acted on only by the Eath's gavity. At any point on its flight path (assuming ai dag to be negligible): 1. its hoizontal component of acceleation is zeo,. its vetical component of acceleation is equal to g, the gavitational field stength at that point. Any pojectile tavels equal hoizontal distances in equal times because its hoizontal component of acceleation is zeo. Its vetical motion is unaffected by its hoizontal motion. The combination of constant hoizontal velocity and constant downwad acceleation leads to a paabolic path. The path can be calculated in two ways: (1) calculating the path step by step o () using the kinematic equations fo constant acceleation. Step by step calculation displacement due to velocity in next moment, if no foce acts vδt change in displacement due to change in velocity δvδt = g(δt) displacement in next time inteval, afte velocity has changed vδt displacement due to velocity at any moment Suppose the magnitude of the velocity at a given moment is v. The displacement in a shot time δt will be vδt, in the diection of the velocity. Daw such a vecto displacement. If thee wee no gavitational foce, the velocity would continue unchanged. The next displacement would again be vδt, continuing on fom the last. Copy the fist displacement vecto and add it to the tip of the fist vecto. Howeve, the velocity does change. If the change in velocity is δv, the change in displacement due to this change in velocity will be δvδt. Its diection, if the acceleation is due to gavity, will be downwads. Since δv = gδt, the downwad change in displacement δvδt = gδtδt = g(δt). Daw in such a vecto, and find the esultant of it and the displacement had thee been no foce. This is the new segment of path, in the next time inteval. Repeat the pocess, stating now with the new segment of path. The change in displacement due to the change in velocity is the same in evey time inteval, if the intevals ae equal. This fact esults in the cuve taking the shape of a paabola. Using kinematic equations If a pojectile is launched hoizontally at speed u, then at time t afte launch, its hoizontal 1 distance fom the launch point x = ut and its vetical distance y = gt (whee the minus sign signifies that the upwads diection has been taken to be positive). Advancing Physics A 9
Pojection at a non-zeo angle above the hoizontal u θ If the pojectile is launched with an initial speed u at angle θ above the hoizontal, its initial hoizontal component of velocity = u cosθ. Because its hoizontal component of acceleation is zeo, then its hoizontal component of velocity is constant. Hence the hoizontal component of its displacement fom launch, at time t afte being launched, x = ut cosθ and its initial vetical component of velocity = u sinθ. Hence: Its vetical component of velocity at time t afte being launched, u y = u sinθ gt. Its vetical component of displacement fom launch, y = ut sin θ ½ gt. The co-odinates (x, y) elative to the launch point ae theefoe x = u t cos θ, y = u t sin θ ½ g t. The highest point is eached at time t 0 when u y = u sin θ g t 0 = 0, hence t 0 = u sin θ / g. The shape of the path is paabolic. The total enegy of a pojectile is constant because the change of kinetic enegy is equal and opposite to the change of potential enegy. A pojectile diected upwads at a non-zeo angle to the vetical loses kinetic enegy which is stoed as potential enegy in the gavitational field, until it eaches its highest point. As it descends, it takes potential enegy fom the field, gaining kinetic enegy. At any point, the kinetic enegy plus potential enegy equals the initial kinetic enegy plus initial potential enegy. Gavitational potential The gavitational potential at a point is the potential enegy pe unit mass of a small object placed at that point. This is the wok done pe unit mass to move a small object fom infinity to that point. The gavitational potential enegy E P of a point mass m is given by E P = m V G, whee V G is the gavitational potential at that point. The SI unit of gavitational potential is J kg 1. Gavitational potential is a scala quantity. An equipotential is a suface of constant potential. No change of potential enegy occus when an object is moved along an equipotential. The lines of foce ae theefoe always pependicula to the equipotential sufaces. The gavitational field stength at a point in a gavitational field is the negative of the potential gadient at that point. In symbols g = dv G / dx. Advancing Physics A 10
In an invese squae gavitational field due to a mass M, the adial component of the field stength is: GM g =. and the gavitational potential is: GM V G = Vaiation of gavitational potential with distance fom the cente of a spheical body 0 distance fom cente / adius of body Vs Vs suface potential Cicula motion An object moving in a hoizontal cicle at constant speed changes its diection of motion continuously. Its velocity is not constant because its diection of motion is not constant. The esultant foce is diected towads the cente of the cicle. It is called the centipetal foce. Fo an object moving at constant speed v along a cicula path of adius, the acceleation towads the cente is: v a = and the centipetal foce F acting on it is: mv F = ma = whee m is the mass of the object. The centipetal foce does no wok on the moving mass because the foce is always at ight angles to the diection of motion. The enegy of the motion is theefoe constant. The time T taken to move once ound the cicula path is T = π / v Advancing Physics A 11
Fo a poof that v a = see Summay Diagams: Centipetal acceleation Keple's laws Keple's laws have to do with the motion of planets in the sola system, and wee explained by Newton using his theoy of gavitation based on an invese squae law fo the gavitational attaction between a planet and the Sun. Keple's thee laws ae as follows: Keple's fist law: each planet moves ound the Sun in an elliptical obit in which the Sun is at one of the focal points of the ellipse. Keple's second law: the line fom the Sun to a planet sweeps out equal aeas in equal times as the planet moves ound its obit. Keple's thid law: the squae of the time peiod of a planet is in popotion to the cube of its mean adius of obit. It can be moe simply expessed fo a cicula obit by saying that the centipetal acceleation v / is equal to the gavitational field GM /, so that the poduct v is constant, equal to GM. A planet continues to move ound the Sun because of the gavitational foce of attaction between it and the Sun. The foce on the planet always acts towads the Sun, slowing the planet down as it moves on its obit away fom the Sun and speeding it up as it appoaches the Sun. Most planets apat fom Mecuy and Pluto move on obits that ae almost cicula. Keple's laws also apply to the motion of comets which move ound the Sun on highly elliptical obits. The fist law descibes the shape of a planetay obit in geneal tems. The Sun is on the majo axis at one of the two focal points of the elliptical obit. Keple s laws A fast Sun S D slow C B elliptical obit Time fom A to B = time fom C to D if aea ABS = aea CDS The second law aose fom Keple's obsevation that the least distance fom Mas to the Sun is 0.9 times the geatest distance. He also obseved that at its geatest distance fom the Sun, the line joining Mas and the Sun tuned at a ate 0.81 times slowe than at its least distance. Conside the line joining the Sun and a planet. Let it tun though a small angle δ θ in time δ t Advancing Physics A 1
as the planet moves along its obit. The planet moves a distance δ θ in time δ t. Going to the limit as δ t tends to zeo, the speed v = d θ / d t. The angula momentum of the planet is m v, and is thus equal to m d θ / d t. But d θ / d t is the aea swept out by the line joining the Sun and planet. Keple's second law is theefoe equivalent to the statement that the angula momentum of a planet is the same at all points on its obit. Poof of Keple's thid law fo the case of a cicula obit speed v planet of mass m Sun = adius cicula obit A planet moving ound a cicula obit o adius in time T has speed π v =. T The foce of gavitational attaction on the planet due to the Sun is GMm F = whee M is the mass of the Sun and m is the mass of the planet. The centipetal acceleation of the planet on its obit ound the Sun is v a =. This must be equal to the gavitational field g = G M / Thus: GM v = which is independent of the mass m of the planet. Substituting v = π T theefoe gives 4π GM =, T Advancing Physics A 13
which can be eaanged to give 4π T = GM 3 in ageement with Keple's thid law. Mecuy Venus Eath Mas Jupite Satun Time peiod in yeas, T (yeas) Mean adius of obit, / mean distance of Eath fom Sun 0.4 0.61 1.0 1.9 11.9 9.6 0.39 0.7 1.0 1.5 5. 9.5 Relationships Keple's second law: dθ dt = constant Keple's thid law: 3 = GM T 4π Satellite motion A satellite is any object in obit about a lage astonomical object. The planets ae satellites of the Sun, the Moon is a satellite of the Eath. Atificial satellites in obit about the Eath ae used fo communications, meteoology and suveillance. The time peiod of a satellite is the time taken to complete one obit. The time peiod of an atificial satellite depends on its distance fom the Eath. A geostationay satellite is a satellite in an equatoial obit at such an altitude and diection that it emains at the same point above the equato because its time peiod is exactly 4 hous. A satellite obiting the Eath is kept in its obit by the foce of gavity between it and the Eath. In geneal, a satellite obit is an ellipse. Fo a cicula obit (a special case of an ellipse), the velocity of the satellite is always pependicula to the foce of gavity on the satellite, and is given by equating the foce of gavitational attaction G M m / to the centipetal foce m v / whee M is the mass of the Eath, m is the mass of the satellite, is the adius of obit and v is the speed of the satellite. Advancing Physics A 14
velocity satellite foce Hence v = G M / gives the speed of a satellite in a cicula obit. The time peiod T = π / v fo a cicula obit. These equations can be used to show that: T = 4 hous fo = 4 300 km. Thus the height of a geostationay obit must be 35 900 km because the Eath's adius is 6400 km. Geostationay satellites ae used fo communications because satellite tansmittes and eceives once pointed towads the satellite emain pointing to it without having to tack it. = 8000 km fo T = hous, thus a satellite in a pola obit with a time peiod of hous needs to be at a height of 1600 km above the Eath. In such an obit, the satellite makes 1 obits evey 4 hous, cossing the equato 30 futhe west on each successive tansit in the same diection. Such satellites ae used fo meteoological and suveillance puposes. The total enegy of a satellite, E = E K + E P, whee E K is its kinetic enegy (= ½ mv ) and E P is its potential enegy ( = GMm / ). Fo a cicula obit, v = GM / so E K = ½ m G M / = ½ E P. Thus the kinetic enegy has magnitude equal to half the potential enegy, and the total enegy E = ½ E P + E P = ½ E P = GMm /. Relationships Fo cicula obits: v = GM /. T = π / v. Advancing Physics A 15
Summay Diagams Back to list of Contents Consevation of momentum Consevation of momentum p = mv Befoe collision: p 1 p [total momentum p] befoe = [m 1 v 1 + m v ] befoe m 1 m Afte collision: p 1 p [total momentum p] afte = [m 1 v 1 + m v ] afte Duing collision: momentum Δp goes fom one mass to the othe befoe: Momentum conseved loses Δp p 1 p Δp Δp [p 1 ] afte = [p 1 ] befoe Δp [p ] afte = [p ] befoe +Δp [Δp] total = 0 gains Δp afte: p 1 p theefoe: [p 1 + p ] afte = [p 1 + p ] befoe Changes of velocity: m 1 Δv 1 = Δp m Δv = +Δp theefoe: Δv Δv 1 = m 1 m changes of momentum ae equal and opposite changes of velocity ae in invese popotion to mass Momentum just goes fom one object to the othe. The total momentum is constant Advancing Physics A 16
Two caft collide Two equally massive spacecaft dock togethe and join. The collision is seen fom two diffeent moving points of view. Momentum is conseved fom both points of view Two cafts appoach one anothe and dock togethe View 1 Obsevation caft hoves whee the caft will meet +v v Pogess Mi obsevation caft video of collision seen fom obsevation caft +v v Pogess Mi View Obsevation caft tavels alongside Mi +v v Pogess Mi v obsevation caft video of collision seen fom obsevation caft +v +v Pogess Mi The same event looks diffeent fom two diffeent points of view Advancing Physics A 17
One event, two points of view Befoe collision +v v Pogess Mi momentum befoe = +mv mv = 0 Afte collision velocity = 0 Pogess Mi momentum afte = 0 Befoe collision velocity of this fame elative to fame above velocity = 0 +v Pogess Mi Afte collision momentum befoe = +m(v) = +mv m +v Pogess Mi momentum afte = (m)v = mv Momentum is diffeent in the two views of the same event, but in each case: momentum afte = momentum befoe Advancing Physics A 18
Moe collisions Hee ae six collisions. Notice that the total momentum befoe is always equal to the total momentum afte. equal masses, inelastic collision befoe velocity v velocity v total momentum befoe 0 duing afte both velocities zeo afte 0 equal masses, elastic collision befoe total momentum velocity v velocity zeo befoe duing afte afte velocity zeo velocity v equal masses, elastic collision befoe velocity v velocity v total momentum befoe 0 duing afte velocity v velocity v afte 0 Advancing Physics A 19
unequal masses, inelastic collision befoe total momentum velocity v velocity zeo befoe duing afte velocity a little less than v afte unequal masses, elastic collision befoe total momentum velocity v velocity zeo befoe duing afte velocity a little less than v velocity much less than v afte unequal masses, elastic collision befoe total momentum velocity v velocity zeo befoe duing afte velocity much less than v velocity up to v afte Advancing Physics A 0
Momentum, invaiance, symmety Thinking about momentum and foces Pinciple 1 symmety +v v identical objects esult pedictable fom symmety Pinciple invaiance +v v seen diffeently is the same as: +v Consevation of momentum Δp +Δp cunch Δp mass M foce F acts fo time Δt change of momentum = F Δt Split cunch into foces F on each foce F = Δp Δt same time Δt. foces equal and opposite foce +F acts fo time Δt mass m +Δp change of momentum = F Δt if define foce F = Δp Δt then F = mδv = ma Δt thus F = ma Fom symmety and invaiance (looking diffeently can t change events): 1. momentum is conseved. define mass fom change of velocity in collision 3. define foce as ate of change of momentum, giving F = ma 4. foces on inteacting objects act in equal and opposite pais Advancing Physics A 1
Jets and ockets Jets and ockets momentum caied by gas plus momentum change of ocket = 0 ocket velocity V inceases by V in time t p momentum caied away by jet: p = v m in time t ocket mass M p change of momentum of ocket: p = M V in time t mass m ejected in time t fo jet: v m = p equal and opposite fo ocket: p = M V M V = v m Rocket thust = v m t gas velocity v thust = V = v m M p = M V = v m t t t Advancing Physics A
Gaphs showing g against h Gavitational field and gavitational potential enegy Unifom gavitational field Field pictue Potential enegy pictue mass m foce mg mass m potential enegy change E h lift by height h 40 J kg 1 30 J kg 1 field stength = g 0 J kg 1 10 J kg 1 1 m foce on mass in gavitational field = mg field = foce/mass = g potential enegy change = foce distance = mg h gavitational potential diffeence = potential change/mass = mg h/m = g h Field is slope of potential hill Potential diffeence is aea unde gaph field g = slope gavitational potential diffeence g aea g h = gavitational potential diffeence h displacement upwads h displacement upwads The field is the ate of change of potential with displacement Advancing Physics A 3
Field and potential Gavitational field and gavitational potential 0 + 0 adius If the potential vaies as 1/ then the field vaies as 1/ Assume 1/ vaiation of potential and calculate diffeence in potential V gav at + at adius : V = GM at adius + : V = GM + V gav at field g = slope V/ V Diffeence in potential V between and + is: ( ) V = GM GM + GM ( + ) + GM V = ( + ) if is small: V = GM field g = V Thus: adial component of field g = GM Radial component of field = dv. Since V = GM, then adial component of field g = GM d d Advancing Physics A 4
Gavitational potential well field down potential slope level in well is potential enegy pe kg Gavitational field and adius 0 0 g V gav = aea g g adius Gavitational potential and adius 0 0 V gav field g = slope V gav / V gav adius The field is the slope of the gaph of potential against adius The diffeence in potential is the aea unde the gaph of field against adius Advancing Physics A 5
Centipetal acceleation Cicula motion cicula path adius speed v A v 1 B v velocity tuns though angle as planet goes along cicula path in shot time t adius tuns though velocity tuns though A speed v v 1 B ac AB v v ac AB = distance in time t at speed v ac AB = v t v t change of velocity v towads cente of cicle v v v t multiply by v: v v t v divide by t: v t v v v = acceleation The acceleation towads the cente of a cicula obit = v Advancing Physics A 6
Geomety ules the Univese Keple: Geomety ules the Univese Law 1: a planet moves in an ellipse w ith the Sun at one focus Astonomy Ma s Geomety planet a b Sun focus focus Obit of Mas is an ellipse with Sun at a focu s Ellipse: cuve such tha t sum of a and b is constan t Law : the line fom the Sun to a planet s weeps out equal aeas in equal times Astonomy Mas Geomety planet fast Sun slow focus Speed of planet faste nea Sun, slo we away fom Sun A eas swept out in same time ae equal Law 3: squae of obital time is popotional to cube of obital adius Obital peiod against obital adius Mas 4 3 Obital p eiod squaed against obital ad ius cub ed Mas 1 Eath Venus 1 Eath Mecuy 0 0 50 100 150 00 50 adius/million km Ve nus 0 Mecuy 0 1 3 4 adius 3 /AU 3 Keple fomulated these thee laws govening the motion of planets aound the sun Advancing Physics A 7
A geostationay satellite A geostationay satellite m = mass of satellite R = adius of satellite obit v = speed in obit G = gavitational constant = 6.67 10 11 N kg m N M = mass of Eath = 5.98 10 4 kg T = time of obit = 4 hous = 86400 s obit adius R = 4000 km gavitational foce S satellite obit tuns at same ate as Eath tuns Calculating the adius of obit foce poducing acceleation to cente mv R equal gavitational foce on satellite GMm R foces ae equal: mv R divide by m: v R = = GMm R GM R speed in obit depends on time of obit and adius R v = T multiply by R: v = GM R equal v 4 R = T equate ex pess ions fo v : GM R = 4 R T eaange to calculate R: GMT 4 = R 3 Keple s thid law deduced inset values of G, M and T: R = 4. 10 4 km R = 6.6 adius of Eath (6400 km) Keple s thid law, and the obit adius of a geostationay satellite, can be deduced fom fist pinciples Advancing Physics A 8
Relationship between g and V g Gavitational gadients aound Eath Equipotentials ae sphees contou of constant gavitational potential gavitational field Advancing Physics A 9
Speeds and acceleations in the Sola System Speeds and acceleations in the Sola System What speed and acceleation? why don t you notice them? Time 1 day = 4 hous Tavelling ound once a day on Eath s equato Distance cicumfeence = 6400 km = 4000 km What speed and acceleation? why don t you notice them? Time 1 yea = 365 days = 8800 hous Tavelling ound the Sun once a yea on Eath s obit Distance cicumfeence = 1.5 10 8 km = 9.4 10 8 km adius 6400 km Acceleation a = v (1700 km pe hou) a = 6400 km a = 440 km h 1 pe hou a = 0.034 m s Speed 4000 km v = 4 hou v = 1700 km pe hou Eath adius 150 million km Sun Acceleation a = v (110000 km pe hou) a = 9.4 10 8 km a = 80 km h 1 pe hou a = 0.006 m s Speed v = 9.4 108 km 8800 hou v = 110000 km pe hou Advancing Physics A 30
Acceleation of the Moon The acceleation of the Moon and the invese squae law Acceleation of the Moon Time 1 Moon month = 7.3 days = 7.3 4 3600 s =.35 10 6 s Tavelling aound the Eath once a month Moon obit adius 384000 km Eath Distance cicumfeence = 3.84 10 8 m = 4.1 10 8 m Diluting Eath s gavity Moon s obit adius = 384000 km Moon diluted gavitational pull obit adius 384000 km Eath g = 9.8 m s at suface Eath s adius adius = 6400 km Ratio of Eath s adius to Moon s obit 6400 km atio = 384000 km = 1 60 Acceleation a = v m a = (100 m s 1 ) 3.84 10 8 m a = 0.007 m s Speed v = 4.1 108 m.35 10 6 s v = 100 m s 1 acceleation found fom motion Gavity diluted by invese squae law Eath s suface: g = 9.8 m s At Moon s obit: 9.8 m s acceleation = 60 a = 0.007 m s same acceleation found fom invese squae law The acceleation of the Moon is simply diluted Eath gavity. The acceleation measues the gavitational field Advancing Physics A 31
Apollo etuns fom the Moon Apollo 11 comes back fom the Moon Pais of obsevations of speed and distance Apollo 11 is coasting home downhill with ockets tuned off. Distances taken fom cente of Eath distance / 10 6 m speed / m s 1 1 v / 10 6 J kg 1 10 8 m / 41.6 09.7 170.9 96.8 56.4 8.4 13.3 151 1676 1915 690 366 501 7673 1.16 1.40 1.83 3.6 6.57 13.53 9.44 0.414 0.477 0.585 1.033 1.774 3.518 7.513 Vaiation of gavitational potential with distance 0 Vaiation of gavitational potential with 1 / 0 10 10 0 1 mv is the kinetic enegy 1 v is the kinetic enegy pe kilogam 1 v is the change in potential enegy pe kilogam 0 30 0 50 100 150 00 distance / 1000 km 50 30 0 4 6 8 10 8 m / Back to list of Contents Advancing Physics A 3