Revision Guide for Chapter 11

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1 Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams (OHTs) Consevation of momentum... 8 Collisions fom diffeent viewpoints... 9 Examples of collisions Momentum and foce Jets and ockets Field diection and equipotentials Centipetal acceleation Satellites and Keple s thid law Advancing Physics A 1

2 Student's 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 as ate of change of momentum consevation of momentum when objects inteact Revision Notes: Momentum; Newton s Laws of motion Summay Diagams: Consevation of momentum; Collisions fom diffeent viewpoints; Examples of collisions; Momentum and foce; 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 and kinetic enegy when objects move in a gavitational field motion in a unifom gavitational field See Revision Guide Chapte 9: Wok; kinetic enegy; potential enegy; fee fall; pojectile the gavitational field and gavitational potential due to a point mass Revision Notes: Gavitational field; Gavitational potential Summay Diagams: Field diection and equipotentials; motion in a hoizontal cicle and in a cicula gavitational obit about a cental mass Revision Notes: Motion in a cicle Summay Diagams: Centipetal acceleation; Keple s laws; Satellites and Keple s thid law I can use the following wods and phases accuately when descibing effects and obsevations: Momentum Revision Notes: Momentum kinetic enegy and potential enegy See Revision Guide Chapte 9: kinetic enegy; potential enegy gavitational field, gavitational potential Advancing Physics A

3 Revision Notes: Gavitational field; Gavitational potential 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 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 diagams illustating gavitational fields and the coesponding equipotential sufaces Summay Diagams: Field diection and equipotentials 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 E = Fs cosθ, (no wok is done when F and s ae pependicula) See Revision Guide Chapte 9: Wok; kinetic enegy; potential enegy; fee fall; pojectile momentum p = mv and F = (mv) / t Revision Notes: Momentum; Newton s Laws of motion cicula and obital motion: a = v /; F = mv / Revision Notes: Motion in a cicle Summay Diagams: Centipetal acceleation; GmM Fgav gavitational fields: F gav =, g = = m GmM gavitational potential enegy E GM gavitational potential V = gav gav = m GM Revision Notes: Gavitational field; Gavitational potential Summay Diagams: Field diection and equipotentials; Advancing Physics A 3

4 Revision Notes Back to list of Contents Momentum Momentum is mass x velocity. Momentum is a vecto quantity. The SI unit of momentum is kg m s 1. Newton's second law defines foce as the ate of change of momentum ( mv) F =. t 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: ( mv) = F t. The poduct F 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 t 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 ate of change of momentum of an object is equal to the esultant foce on the object. 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 dv F = m = ma dt Advancing Physics A 4

5 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 on each. 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. The SI unit of gavitational field stength is N kg -1 o equivalently m s -. The foce F on a point mass m at a point in a gavitational field is given by F = m g, whee g is 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 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 foce acts towads the mass M. The measued value of the Univesal Gavitational Constant G is N m kg -. The gavitational field stength g = F / m = G M / at distance fom the cente of the mass M. Advancing Physics A 5

6 Vaiation of g with distance fom the cente of the Eath distance fom cente adius of Eath 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. In an invese squae gavitational field, the field stength is: GM g =. and the gavitational potential is: GM V G = Advancing Physics A 6

7 Vaiation of gavitational potential with distance fom the cente of a spheical body 0 distance fom cente / adius of body Vs Vs suface potential Motion in a cicle 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 Fo a poof that v a = see Summay diagams: Centipetal acceleation Advancing Physics A 7

8 Summay Diagams (OHTs) 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 p goes fom one mass to the othe befoe: Momentum conseved loses p p 1 p p p [p 1 ] afte = [p 1 ] befoe p [p ] afte = [p ] befoe + p [ p] total = 0 gains p afte: p 1 p theefoe: [p 1 + p ] afte = [p 1 + p ] befoe Changes of velocity: m 1 v 1 = p m v = + p theefoe: v v 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 8

9 Collisions fom diffeent viewpoints 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 9

10 One event seen fom 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 10

11 Examples of 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 11

12 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 1

13 Momentum and foce 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 p + p cunch p mass M foce F acts fo time t change of momentum = F t Split cunch into foces F on each foce F = p t same time t. foces equal and opposite foce +F acts fo time t mass m + p change of momentum = F t if define foce F = p t then F = m v = ma t 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 13

14 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 V = v m M gas velocity v thust = p = M V v m = t t t Advancing Physics A 14

15 Field diection and equipotentials Gavitational gadients aound Eath Equipotentials ae sphees contou of constant gavitational potential gavitational field Advancing Physics A 15

16 Centipetal acceleation Acceleation towads cente of cicula obit cicula path adius speed v A B v 1 v θ velocity tuns though angle θ as planet goes along cicula path in shot time t adius tuns though θ velocity tuns though θ A speed v θ v 1 B v v θ θ = ac AB ac AB = distance in time t at speed v ac AB = v t change of velocity v towads cente of cicle v t = θ θ v v v t v v multiply by v: v v t divide by t: v = = v v t = acceleation Acceleation towads cente = v Advancing Physics A 16

17 Keple: Geomety ules the Univese Law 1: a planet moves in an ellipse with the Sun at one focus Astonomy Mas Geomety planet a b Sun focus focus Obit of Mas an ellipse with Sun at a focus Ellipse: cuve such that sum of a and b is constant Keple: Geomety ules the Univese Law : the line fom the Sun to a planet sweeps out equal aeas in equal times Astonomy Mas Geomety planet fast Sun slow focus Speed of planet lage nea Sun, smalle away fom Sun Aeas swept out in same time ae equal Advancing Physics A 17

18 Keple: Geomety ules the Univese Law 3: squae of obital time is popotional to cube of obital adius Obital peiod against obital adius Mas 4 3 Obital peiod squaed against obital adius cubed Mas 1 Eath Venus 1 Eath Mecuy adius/million km Venus 0 Mecuy adius 3 /AU 3 Advancing Physics A 18

19 Satellites and Keple s thid law The example of a geostationay satellite is used to deive Keple s thid law. Geostationay satellite m = mass of satellite R = adius of satellite obit v = speed in obit G = gavitational constant = N kg m M = mass of Eath = kg T = time of obit = 4 hous = s N 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 v = πr T multiply by R: v = GM R equal v 4π R = T equate expessions fo v : GM R 4π R T eaange to calculate R: GMT 4π = R 3 = Keple s thid law deduced inset values of G, M and T: R = km R = 6.6 adius of Eath (6400 km) Advancing Physics A 19

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