Chapter 10 Angular Momentum

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 10 Angular Momentum"

Transcription

1 Chapte 0 Angula Momentum Conceptual Poblems 5 A paticle tavels in a cicula path and point P is at the cente o the cicle. (a) the paticle s linea momentum p is doubled without changing the adius o the cicle, how is the magnitude o its angula momentum about P aected? (b) the adius o the cicle is doubled but the speed o the paticle is unchanged, how is the magnitude o its angula momentum about P aected? Detemine the Concept and p ae elated accoding to p and the magnitude o is psinφ whee φ is the angle between and p. (a) Because is diectly popotional to p, is doubled. (b) Because is diectly popotional to, is doubled. One way to tell i an egg is hadboiled o uncooked without beaking the egg is to lay the egg lat on a had suace and ty to spin it. A hadboiled egg will spin easily, while an uncooked egg will not. Howeve, once spinning, the uncooked egg will do something unusual; i you stop it with you inge, it may stat spinning again. Explain the dieence in the behavio o the two types o eggs. Detemine the Concept The hadboiled egg is solid inside, so eveything otates with a uniom angula speed. By contast, when you stat an uncooked egg spinning, the yolk will not immediately spin with the shell, and when you stop it om spinning the yolk will continue to spin o a while. 5 The angula momentum o the popelle o a small single-engine aiplane points owad. The popelle otates clockwise i viewed om behind. (a) Just ate lito, as the nose o the plane tilts upwad, the aiplane tends to vee to one side. To which side does it tend to vee and why? (b) the plane is lying hoizontally and suddenly tuns to the ight, does the nose o the plane tend to vee upwad o downwad? Why? (a) The plane tends to vee to the ight. The change in angula momentum Δ pop o the popelle is upwad, so the net toque τ on the popelle is upwad as well. The popelle must exet an equal but opposite toque on the plane. This downwad toque exeted on the plane by the popelle tends to cause a downwad change in the angula momentum o the plane. This means the plane tends to otate clockwise as viewed om above. 95

2 96 Chapte 0 (b) The nose o the plane tends to vee downwad. The change in angula momentum Δ pop o the popelle is to the ight, so the net toque τ on the popelle is towad the ight as well. The popelle must exet an equal but opposite toque on the plane. This letwad diected toque exeted by the popelle on the plane tends to cause a letwad-diected change in angula momentum o the plane. This means the plane tends to otate clockwise as viewed om the ight. 7 You ae sitting on a spinning piano stool with you ams olded. (a) When you extend you ams out to the side, what happens to you kinetic enegy? What is the cause o this change? (b) Explain what happens to you moment o inetia, angula speed and angula momentum as you extend you ams. Detemine the Concept The otational kinetic enegy o the you-stool system is given by K ot ω. Because the net toque acting on the you-stool system is zeo, its angula momentum is conseved. (a) You kinetic enegy deceases. nceasing you moment o inetia while conseving you angula momentum deceases you kinetic enegy K. (b) Extending you ams out to the side inceases you moment o inetia and deceases you angula speed. The angula momentum o the system is unchanged. Estimation and Appoximation 9 An ice-skate stats he piouette with ams outstetched, otating at.5 ev/s. Estimate he otational speed (in evolutions pe second) when she bings he ams lat against he body. Pictue the Poblem Because we have no inomation egading the mass o the skate, we ll assume that he body mass (not including he ams) is 50 kg and that each am has a mass o 4.0 kg. et s also assume that he ams ae.0 m long and that he body is cylindical with a adius o 0. Because the net extenal toque acting on he is zeo, he angula momentum will emain constant duing he piouette. Because the net extenal toque acting on he is zeo: Δ i 0 o ω ω 0 () ams in amsin amsout amsout

3 Angula Momentum 97 Expess he total moment o inetia with he ams outstetched: Teating he body as though it is cylindical, calculate the moment o inetia o he body, minus he ams: Modeling he ams as though they ae ods, calculate thei moment o inetia when they ae outstetched: + ams out body ams m body.00 kg m ams ( 50kg)( 0.0m) [ ( 4kg)(.0m) ].67 kg m Substitute to detemine he total moment o inetia with he ams outstetched: Expess he total moment o inetia with he ams lat against he body: ams out ams in.00 kg m.67 kg m + body + ams.00 kg m. kg m +.67 kg m [( 4.0kg)( 0.0m) ] Solve equation () o ω ams in to obtain: amsout ω ams in ω amsin amsout Substitute numeical values and evaluate ω : ams in ω ams in.67 kg m. kg m 4ev/s (.5ev/s) A.0-g paticle moves at a constant speed o.0 mm/s aound a cicle o adius 4.0 mm. (a) Find the magnitude o the angula momentum o the paticle. (b) l( l + )h, whee l is an intege, ind the value o l( l+) and the appoximate value o l. (c) By how much does l change i the paticle s speed inceases by one-millionth o a pecent, and nothing else changing? Use you esult to explain why the quantization o angula momentum is not noticed in macoscopic physics. Pictue the Poblem We can use mv to ind the angula momentum o the paticle. n (b) we can solve the equation l( l + )h o l( l +) and the appoximate value o l.

4 98 Chapte 0 (a) Use the deinition o angula momentum to obtain: mv.40 0 (.0 0 kg)(.0 0 m/s)( m) kg m /s.4 0 kg m /s (b) Solve the equation l( l + )h o l ( l +) : l + () h ( l ) Substitute numeical values and l + evaluate l ( ): l( l ).40 0 kg m /s.05 0 J s Because l >>, appoximate its value with the squae oot o l l + : ( ) l. 0 6 (c) The change in l is: the paticle s speed inceases by one-millionth o a pecent while nothing else changes: Δ l l l () new v v + 0 and + 0 v ( + 0 )v ( + 0 ) Equation () becomes: l new and l new ( ) [( + 0 ) ] l + new ( + 0 ) Substituting in equation () yields: ( + 0 ) h Δl l new l 0 h h h h Substitute numeical values and evaluate Δ l : Δl.40 0 kg m /s.05 0 J s and 8 l l. 0 Δ 6 %

5 Angula Momentum 99 The quantization o angula momentum is not noticed in macoscopic physics because no expeiment can detect a actional change in l o 0 6 %. The Coss Poduct and the Vecto Natue o Toque and Rotation 7 A oce o magnitude F is applied hoizontally in the negative x diection to the im o a disk o adius R as shown in Figue 0-4. Wite F and in tems o the unit vectos ˆ i, j ˆ, and k ˆ, and compute the toque poduced by this oce about the oigin at the cente o the disk. Pictue the Poblem We can expess F and in tems o the unit vectos î and ĵ and then use the deinition o the coss poduct to ind τ. Expess F in tems o F and the unit vecto : î F Fiˆ Expess in tems o R and the unit vecto : ĵ Rˆj ( ) ( ) F τ F FR ˆj iˆ FR iˆ ˆj Calculate the coss poduct o and : Toque and Angula Momentum FR kˆ 7 A.0-kg paticle moves diectly eastwad at a constant speed o 4.5 m/s along an east-west line. (a) What is its angula momentum (including diection) about a point that lies 6.0 m noth o the line? (b) What is its angula momentum (including diection) about a point that lies 6.0 m south o the line? (c) What is its angula momentum (including diection) about a point that lies 6.0 m diectly east o the paticle? Pictue the Poblem The angula momentum o the paticle is p whee is the vecto locating the paticle elative to the eeence point and p is the paticle s linea momentum. (a) The magnitude o the paticle s angula momentum is given by: Substitute numeical values and evaluate : psinφ mvsinφ mv (.0kg)( 4.5m/s)( 6.0m) 54kg m /s ( sinφ)

6 00 Chapte 0 Use a ight-hand ule to establish the diection o : 54kg m /s, upwad (b) Because the distance to the line along which the paticle is moving is the same, only the diection o dies: (c) Because p 0 o a point on the line along which the paticle is moving: 54kg m /s, downwad 0 45 n Figue 0-46, the incline is ictionless and the sting passes though the cente o mass o each block. The pulley has a moment o inetia and adius R. (a) Find the net toque acting on the system (the two masses, sting, and pulley) about the cente o the pulley. (b) Wite an expession o the total angula momentum o the system about the cente o the pulley. Assume the masses ae moving with a speed v. (c) Find the acceleation o the masses by using you esults o Pats (a) and (b) and by setting the net toque equal to the ate o change o the system s angula momentum. Pictue the Poblem et the system include the pulley, sting, and the blocks and assume that the mass o the sting is negligible. The angula momentum o this system changes because a net toque acts on it. (a) Expess the net toque about the cente o mass o the pulley: Rg( m sinθ m ) (b) Expess the total angula momentum o the system about an axis though the cente o the pulley: τ net Rm g sinθ Rm g whee we have taken clockwise to be pos to be consistent with a positive upwad velocity o the block whose mass is m as indicated in the igue. ω + m vr + m vr vr R + m + m (c) Expess τ as the time deivative o the angula momentum: d τ dt ar R d dt vr R + m + m + m + m

7 Equate this esult to that o Pat (a) and solve o a to obtain: a Angula Momentum 0 ( m θ m ) g R sin + m + m Consevation o Angula Momentum 49 You stand on a ictionless platom that is otating at an angula speed o.5 ev/s. You ams ae outstetched, and you hold a heavy weight in each hand. The moment o inetia o you, the extended weights, and the platom is 6.0 kg m. When you pull the weights in towad you body, the moment o inetia deceases to.8 kg m. (a) What is the esulting angula speed o the platom? (b) What is the change in kinetic enegy o the system? (c) Whee did this incease in enegy come om? Pictue the Poblem et the system consist o you, the extended weights, and the platom. Because the net extenal toque acting on this system is zeo, its angula momentum emains constant duing the pulling in o the weights. (a) Using consevation o angula momentum, elate the initial and inal angula speeds o the system to its initial and inal moments o inetia: Substitute numeical values and evaluate ω : (b) Expess the change in the kinetic enegy o the system: i i ωi ω 0 ω ωi 6.0kg m ω.8kg m Δ K K K i (.5ev/s) 5.0ev/s ω ω i i Substitute numeical values and evaluate ΔK: ΔK ev π ad (.8kg m ) 5.0 ( 6.0kg m ) 0.6 kj s ev ev π ad.5 s ev (c) Because no extenal agent does wok on the system, the enegy comes om you intenal enegy.

8 0 Chapte 0 5 A lazy Susan consists o a heavy plastic disk mounted on a ictionless beaing esting on a vetical shat though its cente. The cylinde has a adius R 5 and mass M 0.5 kg. A cockoach (mass m 0.05 kg) is on the lazy Susan, at a distance o 8.0 om the cente. Both the cockoach and the lazy Susan ae initially at est. The cockoach then walks along a cicula path concentic with the cente o the azy Susan at a constant distance o 8.0 om the axis o the shat. the speed o the cockoach with espect to the lazy Susan is 0.00 m/s, what is the speed o the cockoach with espect to the oom? Pictue the Poblem Because the net extenal toque acting on the lazy Susancockoach system is zeo, the net angula momentum o the system is constant (equal to zeo because the lazy Susan is initially at est) and we can use consevation o angula momentum to ind the angula velocity ω o the lazy Susan. The speed o the cockoach elative to the loo v is the dieence between its speed with espect to the lazy Susan and the speed o the lazy Susan at the location o the cockoach with espect to the loo. Relate the speed o the cockoach with espect to the loo v to the speed o the lazy Susan at the location o the cockoach: Use consevation o angula momentum to obtain: Expess the angula momentum o the lazy Susan: Expess the angula momentum o the cockoach: Substitute o S and C in equation () to obtain: Solving o ω yields: v v ω () 0 () S C ω S C S MR ω v CωC m ω v MR ω m ω ω mv MR + m 0 Substitute o ω in equation () to m v obtain: v v MR + m Substitute numeical values and evaluate v : ( 0.05kg)( m) ( 0.00 m/s) ( 0.5m)( 0.5m) + ( 0.05kg)( m) v 0.00 m/s 0 mm/s

9 Angula Momentum 0 Remaks: Because the moment o inetia o the lazy Susan is so much lage than the moment o inetia o the cockoach, ate the cockoach begins moving, the angula speed o the lazy Susan is vey small. Theeoe, the speed o the cockoach elative to the loo is almost the same as the speed elative to the lazy Susan. *Quantization o Angula Momentum 55 The z component o the spin o an electon is h, but the magnitude o the spin vecto is 0.75h. What is the angle between the electon s spin angula momentum vecto and the positive z-axis? Pictue the Poblem The electon s spin angula momentum vecto is elated to its z component as shown in the diagam. The angle between s and the positive z-axis is φ. z h φ θ 0.75h s h Expess φ in tems o θ to obtain: φ 80 θ Using tigonomety, elate the magnitude o s to its z component: h θ cos 0.75h Substitute o θ in the expession o φ to obtain: h θ 80 cos 0.75h 5 57 You wok in a bio-chemical eseach lab, whee you ae investigating the otational enegy levels o the HB molecule. Ate consulting the peiodic chat, you know that the mass o the bomine atom is 80 times that o the hydogen atom. Consequently, in calculating the otational motion o the molecule, you assume, to a good appoximation, that the B nucleus emains stationay as the H atom (mass kg) evolves aound it. You also know that the sepaation between the H atom and bomine nucleus is 0.44 nm. Calculate (a) the moment o inetia o the HB molecule about the bomine nucleus, and (b) the otational enegies o the bomine nucleus s gound state (lowest enegy) l 0, and the next two states o highe enegy (called the ist and second excited states) descibed by l, and l.

10 04 Chapte 0 Pictue the Poblem The otational enegies o HB molecule ae elated to l and E0 accoding to K l l( l +) E 0 whee E0 h. (a) Neglecting the motion o the bomine molecule: Substitute numeical values and evaluate HB : HB mp HB m H 7 9 (.67 0 kg)( m) kg m kg m (b) Relate the otational enegies to l and E : 0 ( ) K l l l + E 0 whee E 0 h HB Substitute numeical values and evaluate E : 0 E 0 h meV 4 ( J s) 47 ( 0 kg m ).46 ev J J Evaluate E 0 to obtain: E K meV Evaluate E to obtain: E K ( + )(.00meV).0meV Evaluate E to obtain: E K ( + )(.00meV) Collisions with Rotations 6.0meV 6 Figue 0-5 shows a thin uniom ba o length and mass M and a small blob o putty o mass m. The system is suppoted by a ictionless hoizontal suace. The putty moves to the ight with velocity v, stikes the ba at a distance d om the cente o the ba, and sticks to the ba at the point o contact. Obtain expessions o the velocity o the system s cente o mass and o the angula speed ollowing the collision. Pictue the Poblem The velocity o the cente o mass o the ba-blob system does not change duing the collision and so we can calculate it beoe the collision

11 Angula Momentum 05 using its deinition. Because thee ae no extenal oces o toques acting on the ba-blob system, both linea and angula momentum ae conseved in the collision. et the diection the blob o putty is moving initially be the +x diection. et lowe-case lettes ee to the blob o putty and uppe-case lettes ee to the ba. The diagam to the let shows the blob o putty appoaching the ba and the diagam to the ight shows the ba-blob system otating about its cente o mass and tanslating ate the peectly inelastic collision. M M d y d v m v h ω The velocity o the cente o mass beoe the collision is given by: Using its deinition, expess the location o the cente o mass elative to the cente o the ba: Expess the angula momentum, elative to the cente o mass, o the ba-blob system: Expess the angula momentum about the cente o mass: ( M m) v mv + MV + o, because V 0, m v v M + m md ( M + m) y md y M + m below the cente o the ba. ω ω () mv ( d y ) md mmvd mv d M + m M + m

12 06 Chapte 0 Using the paallel axis theoem, expess the moment o inetia o the system elative to its cente o mass: M + My + m d ( y ) Substitute o y and simpliy to obtain: md md M + M + m d M + m M + m Substitute o and in equation ω mmd M + M + m mmvd M M + m + Mmd () and simpliy to obtain: ( ) Remaks: You can veiy the expession o by letting m 0 to obtain M and letting M 0 to obtain A uniom od o length and mass M equal to 0.75 kg is attached to a hinge o negligible mass at one end and is ee to otate in the vetical plane (Figue 0-55). The od is eleased om est in the position shown. A paticle o mass m 0.50 kg is suppoted by a thin sting o length om the hinge. The paticle sticks to the od on contact. What should the atio / be so that θ max 60 ate the collision? Pictue the Poblem Assume that thee is no iction between the od and the hinge. Because the net extenal toque acting on the system is zeo, angula momentum is conseved in this peectly inelastic collision. The od, on its downwad swing, acquies otational kinetic enegy. Angula momentum is conseved in the peectly inelastic collision with the paticle and the otational kinetic o the ate-collision system is then tansomed into gavitational potential enegy as the od-plus-paticle swing upwad. et the zeo o gavitational potential enegy be at a distance below the pivot and use both angula momentum and mechanical enegy consevation to elate the distances and and the masses M and m. Use consevation o enegy to elate the initial and inal potential enegy o the od to its otational kinetic enegy just beoe it collides with the paticle: K Ki + U U i o, because K i 0, K + U U 0 i 0

13 Angula Momentum 07 Substitute o K, U, and U i to obtain: ( M ) ω + Mg Mg 0 Solving o ω yields: ω g etting ω epesent the angula speed o the od-and-paticle system just ate impact, use consevation o angula momentum to elate the angula momenta beoe and ate the collision: Solve o ω to obtain: Use consevation o enegy to elate the otational kinetic enegy o the od-plus-paticle just ate thei collision to thei potential enegy when they have swung though an angle θ max : Δ i 0 o M + m ω' M ω ( ) ( ) 0 ω' M + m K M ω Ki + U U i 0 Because K 0: ( )( cosθ ) + mg ( cosθ ) 0 ω' + Mg () max max Expess the moment o inetia o the system with espect to the pivot: M + m Substitute o θ max, and ω in equation (): g ( M ) M + m Mg ( ) + mg Simpliy to obtain: m m m M M M

14 08 Chapte 0 Dividing both sides o the equation by yields: m 6 + m m + M M M et α m/m and β / to obtain: 6α β + αβ + αβ 0 Substitute o α and simpliy to obtain the cubic equation in β: 8β + 6β + 4β 0 Use the solve unction o you calculato to ind the only eal value o β: Pecession β A bicycle wheel that has a adius equal to 8 is mounted at the middle o an axle 50 long. The tie and im weigh 0 N. The wheel is spun at ev/s, and the axle is then placed in a hoizontal position with one end esting on a pivot. (a) What is the angula momentum due to the spinning o the wheel? (Teat the wheel as a hoop.) (b) What is the angula velocity o pecession? (c) How long does it take o the axle to swing though 60 aound the pivot? (d) What is the angula momentum associated with the motion o the cente o mass, that is, due to the pecession? n what diection is this angula momentum? Pictue the Poblem We can detemine the angula momentum o the wheel and the angula velocity o its pecession om thei deinitions. The peiod o the pecessional motion can be ound om its angula velocity and the angula momentum associated with the motion o the cente o mass om its deinition. (a) Using the deinition o angula momentum, expess the angula momentum o the spinning wheel: w ω MR ω R ω g Substitute numeical values and 0 N 9.8m/s ev π ad s ev 8.J s 8J s ( 0.8m) evaluate :

15 Angula Momentum 09 (b) Using its deinition, expess the angula velocity o pecession: Substitute numeical values and evaluate ω p : d φ MgD ω p dt ω p ( 0 N)( 0.5m) 8.J s 0.4ad/s 0.44 ad/s (c) Expess the peiod o the pecessional motion as a unction o the angula velocity o pecession: (d) Expess the angula momentum o the cente o mass due to the pecession: π π T ω 0.44 ad/s p p ω MD ω p p 5s Substitute numeical values and evaluate p 0 N 9.8m/s J s : p ( 0.5m) ( 0.44 ad/s) Geneal Poblems The diection o p is eithe up o down, depending on the diection o. 7 A paticle whose mass is.0 kg moves in the xy plane with velocity v (.0 m / s)ˆ i along the line y 5. m. (a) Find the angula momentum about the oigin when the paticle is at ( m, 5. m). (b) A oce F (.9 N) ˆ i is applied to the paticle. Find the toque about the oigin due to this oce as the paticle passes though the point ( m, 5. m). Pictue the Poblem While the -kg paticle is moving in a staight line, it has angula momentum given by p whee is its position vecto and p is its linea momentum. The toque due to the applied oce is given by τ F. (a) The angula momentum o the paticle is given by: p

16 0 Chapte 0 m ˆ + 5.m Expess the vectos and p : ( ) i ( )j and p mviˆ.0kg ( 9.0kg m/s)iˆ ( )(.0 m/s) ˆ iˆ Substitute o and p :and simpliy to ind : (b) Using its deinition, expess the toque due to the oce: Substitute o and F and simpliy to ind τ : [( m) iˆ + ( 5.m) ˆj ] ( 9.0kg m/s) ( 47.7 kg m /s)( ˆj iˆ ) τ F τ ( 48kg m /s)kˆ [( m) iˆ + ( 5.m) ˆj ] (.9 N) ( 5.9 N m)( ˆj iˆ ) ( N m)kˆ iˆ iˆ 77 Repeat Poblem 76, this time iction between the disks and the walls o the cylinde is not negligible. Howeve, the coeicient o iction is not geat enough to pevent the disks om eaching the ends o the cylinde. Can the inal kinetic enegy o the system be detemined without knowing the coeicient o kinetic iction? Detemine the Concept Yes. The solution depends only upon consevation o angula momentum o the system, so it depends only upon the initial and inal moments o inetia. 79 Keple s second law states: The line om the cente o the Sun to the cente o a planet sweeps out equal aeas in equal times. Show that this law ollows diectly om the law o consevation o angula momentum and the act that the oce o gavitational attaction between a planet and the Sun acts along the line joining the centes o the two celestial objects. Pictue the Poblem The pictoial epesentation shows an elliptical obit. The tiangula element o the aea is da ( dθ ) d θ.

17 Angula Momentum dθ da θ Dieentiate da with espect to t to obtain: Because the gavitational oce acts along the line joining the two objects, τ 0. Hence: θ da d ω () dt dt m ω constant () Eliminate ω between equations () and () to obtain: da dt m constant 8 The tem pecession o the equinoxes ees to the act that Eath s spin axis does not stay ixed but sweeps out a cone once evey 6,000 y. (This explains why ou pole sta, Polais, will not emain the pole sta oeve.) The eason o this instability is that Eath is a giant gyoscope. The spin axis o Eath pecesses because o the toques exeted on it by the gavitational oces o the Sun and moon. The angle between the diection o Eath s spin axis and the nomal to the ecliptic plane (the plane o Eath s obit) is.5 degees. Calculate an appoximate value o this toque, given that the peiod o otation o Eath is.00 d and its moment o inetia is kg m. Deleted: the eath Pictue the Poblem et ω P be the angula velocity o pecession o Eath-asgyoscope, ω s its angula velocity about its spin axis, and its moment o inetia with espect to an axis though its poles, and elate ω P to ω s and using its deinition. Use its deinition to expess the pecession ate o Eath as a giant gyoscope: Substitute o and solve o τ to obtain: τ ω P τ ω P ωω P

18 Chapte 0 The angula velocity ω s o Eath about its spin axis is given by: Substitute o ω to obtain: ω π whee T is the peiod o T otation o Eath. π ωp τ T Substitute numeical values and evaluateτ: τ π 7 ( kg m )( s ) N m 4h 600s d d h

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

Exam 3: Equation Summary

Exam 3: Equation Summary MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

Multiple choice questions [60 points]

Multiple choice questions [60 points] 1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

More information

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43 Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Revision Guide for Chapter 11

Revision Guide for Chapter 11 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

More information

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2  2 = GM . Combining the results we get ! Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

More information

rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum

rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum Final Exam Duing class (1-3:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday

More information

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

More information

Chapter 9 Rotation. Conceptual Problems

Chapter 9 Rotation. Conceptual Problems Chapte 9 otation Conceptual Poblems 7 Duing a baseball game, the pitche has a blazing astball. You have not been able to swing the bat in time to hit the ball. You ae now just tying to make the bat contact

More information

Phys 2101 Gabriela González. cos. sin. sin

Phys 2101 Gabriela González. cos. sin. sin 1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe

More information

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere. Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

More information

Gravitation. AP Physics C

Gravitation. AP Physics C Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

More information

Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2 F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

More information

Physics 107 HOMEWORK ASSIGNMENT #14

Physics 107 HOMEWORK ASSIGNMENT #14 Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The

More information

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013 PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

More information

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013 PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

A) 2 B) 2 C) 2 2 D) 4 E) 8

A) 2 B) 2 C) 2 2 D) 4 E) 8 Page 1 of 8 CTGavity-1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between

More information

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION

TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................

More information

14. Gravitation Universal Law of Gravitation (Newton):

14. Gravitation Universal Law of Gravitation (Newton): 14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1

More information

Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algebra and Trig. I. A point is a location or position that has no size or dimension. Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

More information

EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD

EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD 260 16-1. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and

More information

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Universal Gravitation 8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

More information

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of universal gravitation Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information

7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary

7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary 7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

More information

XIIth PHYSICS (C2, G2, C, G) Solution

XIIth PHYSICS (C2, G2, C, G) Solution XIIth PHYSICS (C, G, C, G) -6- Solution. A 5 W, 0 V bulb and a 00 W, 0 V bulb ae connected in paallel acoss a 0 V line nly 00 watt bulb will fuse nly 5 watt bulb will fuse Both bulbs will fuse None of

More information

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

More information

Lab M4: The Torsional Pendulum and Moment of Inertia

Lab M4: The Torsional Pendulum and Moment of Inertia M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding

More information

(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

More information

Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics

Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics 3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gap-system.og/~histoy/mathematicians/ Newton.html http://www.fg-a.com http://www.clke.com/clipat

More information

PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013 PHYSICS 111 HOMEWORK SOLUTION #10 April 10, 013 0.1 Given M = 4 i + j 3 k and N = i j 5 k, calculate the vector product M N. By simply following the rules of the cross product: i i = j j = k k = 0 i j

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions

Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions Physics 111 Fall 007 Electostatic Foces an the Electic Fiel - Solutions 1. Two point chages, 5 µc an -8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the

More information

Introduction to Fluid Mechanics

Introduction to Fluid Mechanics Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body

More information

Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

More information

Chapter F. Magnetism. Blinn College - Physics Terry Honan

Chapter F. Magnetism. Blinn College - Physics Terry Honan Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.

More information

Determining solar characteristics using planetary data

Determining solar characteristics using planetary data Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

More information

F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

F G r. Don't confuse G with g: Big G and little g are totally different things. G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

More information

Solutions for Physics 1301 Course Review (Problems 10 through 18)

Solutions for Physics 1301 Course Review (Problems 10 through 18) Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal

More information

Solution Derivations for Capa #8

Solution Derivations for Capa #8 Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

More information

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

More information

10. Collisions. Before During After

10. Collisions. Before During After 10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

More information

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds. Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

AP Physics Electromagnetic Wrap Up

AP Physics Electromagnetic Wrap Up AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle

More information

ESCAPE VELOCITY EXAMPLES

ESCAPE VELOCITY EXAMPLES ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs

More information

Chapter 13. Vector-Valued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates

Chapter 13. Vector-Valued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates 13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. Vecto-Valued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

More information

Magnetism: a new force!

Magnetism: a new force! -1 Magnetism: a new foce! o fa, we'e leaned about two foces: gaity and the electic field foce. F E = E, FE = E Definition of E-field kq E-fields ae ceated by chages: E = 2 E-field exets a foce on othe

More information

Deflection of Electrons by Electric and Magnetic Fields

Deflection of Electrons by Electric and Magnetic Fields Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

More information

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleration Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

Multiple choice questions [70 points]

Multiple choice questions [70 points] Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions

More information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

Chapter 30: Magnetic Fields Due to Currents

Chapter 30: Magnetic Fields Due to Currents d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

More information

Notes on Electric Fields of Continuous Charge Distributions

Notes on Electric Fields of Continuous Charge Distributions Notes on Electic Fields of Continuous Chage Distibutions Fo discete point-like electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution

More information

Fluids Lecture 15 Notes

Fluids Lecture 15 Notes Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

More information

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1

2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation

More information

In this section we shall look at the motion of a projectile MOTION IN FIELDS 9.1 PROJECTILE MOTION PROJECTILE MOTION

In this section we shall look at the motion of a projectile MOTION IN FIELDS 9.1 PROJECTILE MOTION PROJECTILE MOTION MOTION IN FIELDS MOTION IN FIELDS 9 9. Pojectile motion 9. Gavitational field, potential and enegy 9.3 Electic field, potential and enegy 9. PROJECTILE MOTION 9.. State the independence of the vetical

More information

Problems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)

Problems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968) Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems

More information

2. Orbital dynamics and tides

2. Orbital dynamics and tides 2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

More information

Ch. 14: Gravitation (Beta Version 7/01) 14 Gravitation

Ch. 14: Gravitation (Beta Version 7/01) 14 Gravitation Ch. 14: Gavitation (Beta Vesion 7/01) 14 Gavitation The Milky Way galaxy is a disk-shaped collection of dust, planets, and billions of stas, including ou Sun and sola system. The foce that binds it o any

More information

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z. Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm

More information

6.2 Orbits and Kepler s Laws

6.2 Orbits and Kepler s Laws Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that

More information

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

More information

Newton s Shell Theorem

Newton s Shell Theorem Newton Shell Theoem Abtact One of the pincipal eaon Iaac Newton wa motivated to invent the Calculu wa to how that in applying hi Law of Univeal Gavitation to pheically-ymmetic maive bodie (like planet,

More information

Physics 1A Lecture 10C

Physics 1A Lecture 10C Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

More information

Chapter 2. Electrostatics

Chapter 2. Electrostatics Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

More information

Introduction to Electric Potential

Introduction to Electric Potential Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic

More information

Charges, Coulomb s Law, and Electric Fields

Charges, Coulomb s Law, and Electric Fields Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

Gauss Law in dielectrics

Gauss Law in dielectrics Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics,

More information

Stokes law. Viscosity coefficient

Stokes law. Viscosity coefficient EXERCISE 7 Stokes law. Viscosity coeicient 7.1. Intoduction Real luid has a cetain amount o intenal iction, which is called viscosity. Viscosity exists in both liquids and ases, and is essentially the

More information

Gravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.

Gravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C. Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law

More information

CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pre-Calculus Section 10.7. Polar Coordinates CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information