The gravitational force law: Application

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PHYS 7: Moden Mechanic Sping 0 VERY IMPORTANT: Scoe fo lectue -5 quizze wee impoted into CHIP.

o enteed them incoectly Pleae check you coe a..a.p.

CHIP, o enteed wong ID. Pleae eente you ID and notify you lectue immediately.

1 PHYS 7: Moden Mechanic Sping 0 VERY IMPORTANT: Scoe fo lectue -5 quizze wee impoted into CHIP. Moe than 00 tudent did not ente clicke ID into chip o enteed them incoectly Pleae check you coe a..a.p. If you ued iclicke in lectue but you coe doe not appea in CHIP, chance ae that have not egiteed you iclicke ID with CHIP, o enteed wong ID. Pleae eente you ID and notify you lectue immediately. No complaint will be conideed afte lectue #8. The failue to do that by today date may affect you Roll Call tate. Lectue 6: Gavity, Iteative pediction, Electical foce, Uncetainty Read The gavitational foce law: Application m ˆ m m m = G ˆ gav on by Newton G = Cavendih N m kg Gavitational contant m m

2 m m = G ˆ gav on by Gavitational foce on a planet ta 30 m = 4 0 kg =,,.5 0 m. Calculate =,.5, 0 m planet 4 m = 3 0 kg = 3, 3.5, m. Ditance = ( 0 m) + (.5 0 m) + ( 0 m) = m,.5, 0 m 3. Unit vecto: ˆ = = = 0.99, 0.746, m 3. oce: m m G gav on planet byta = ,-0.746,0.597 N magnitude diection ˆ gav on by = = ,0.746, N m m = G ˆ gav on by Gavitational foce on a planet ta 30 m = 4 0 kg =,,.5 0 m = planet 4 m = 3 0 kg = 3, 3.5, m gav on planet byta ,-0.746,0.597 N Checking eult:. Diagam. Ode of magnitude 3. Unit 4. Unit vecto Clicke quetion # : What i the gavitational foce exeted by the planet on the ta? A) The ame B) gav on ta by planet = ,0.746, N gav on ta by planet ,0.746, N C) =

Gavitational foce nea the Eath uface m M m G E R E ˆ gav on mby M = E R E M E = 5.976 0 4 kg R E = 6.

3 Gavitational foce nea the Eath uface m M m G E R E ˆ gav on mby M = E R E M E = kg R E = m Gavitational field ~ The ame fo all object on uface gav on mby M E E = gm M g = G E R ˆ The magnitude: g = 9.8 N/kg g = mg Pedicting motion of a planet Whee will the planet be afte one month? p Ue poition update fomula: = + v t f i avg If we aume that velocity i contant Doe not wok becaue the foce i changing the velocity! The foce change with poition. The momentum change with poition. In geneal, thee i no algebaic equation to pedict motion of moe than inteacting object.

4 Iteative pediction of a motion of one planet p Simple cae: one planet ta i fixed in pace. Calculate gavitational foce: mm ˆ gav on by = G p = p + t. Update momentum f i net Chooe t hot enough ( & v do not change much) 3. Calculate v and update poition 4. Repeat = + v t f i avg Citical paamete: t Iteative pediction of motion Real cae: many object object ae fee to move. Calculate net foce on each ma: on mi = i j m j on mi. Update momentum of each ma p = p + t f i net Chooe t hot enough ( & v do not change much) Iteative appoach: wok fo any kind of foce, not jut gavity! 3. Calculate v and update poition of each ma 4. Repeat = + v t f i avg t i a citical paamete!

5 Iteative pediction of motion: thowing a ock Contant oce (mg) + Momentum Pinciple = Pojectile Motion (cuved path) But, can alo add ai eitance = non-contant foce Example: ma on ping, equilibium How fa will the ping tetch? k Equilibium: momentum doe not change p t = net mg 0 = net 0 = 0, k mg,0 = k mg = mg / k 0

6 Example: ma on ping, in motion mg k mg k mg A ping tetche, foce get lage. Electic foce: the electic chage Chage: popety of an object Two type: poitive (+) and negative (-) Like chage: epel. Oppoite chage: attact Net chage of a ytem: algebaic um of all the chage Conevation of chage The foce exeted by one point chage on anothe act along the line joining the chage Chage: meaued in C (Coulomb) Elementay chage: e = C Chage of electon i e, of a poton +e

The electic foce law (Coulomb law) q elec onby elec q on by ˆ Coulomb law q q = ˆ elec on by 4πε 0 9 N m 9 0 4πε = C 0 Electic foce veu gavity Gavity (Cavendih, 798) mm g

7 The electic foce law (Coulomb law) q elec onby elec q on by ˆ Coulomb law q q = ˆ elec on by 4πε 0 9 N m 9 0 4πε = C 0 Electic foce veu gavity Gavity (Cavendih, 798) mm g = G G = m 3 /(kg. ) Electic foce (Coulomb, 795) q q e = 4πε 0 /(4πε 0 ) = Nm /C 3 m 3 m m = m 70kg = 8 q = q 0 e = C g = N e = N

Pedicting the futue of a gavitational ytem fixed Maive

Deteminim: If we know the poition and momenta of all

Binay ta Sun, Eath and Moon Poblem: Senitivity Initial

8 Pedicting the futue of a gavitational ytem fixed Maive ta And mall planet Two body: ellipe (o cicle) fixed Deteminim: If we know the poition and momenta of all paticle in the Univee we can pedict the futue I thee fee will? Pedicting the futue of gavitational ytem Sola ytem Binay ta Sun, Eath and Moon Poblem: Senitivity Initial condition and t Inability to account fo all inteaction! 0 5 molecule in gla of wate! Small paticle: quantum mechanic

9 Pobability and uncetainty + Example: a fee neuton decay with ~5 minute: n p + e + ν Pobability t Clicke: Can we pedict the motion of an electon nea a fee neuton? A)Ye B) No

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