Newtonian mechanics
Newtonian mechanics
Newtonian mechanics
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Planetbevægelser<br />
Potentiel energi<br />
4 FORMELSAMLING 20<br />
1. Acceleration radial: ar = ¨r − rω 2 = − h2<br />
p<br />
2. Acceleration tangential: aθ = 2 ˙rω − r ˙ω = 0<br />
3. Konstant gravitationsfelt: U(h) = mgh<br />
4. Kepler 3: a3<br />
T 2 = C<br />
5. Sidereal periode T = 2A<br />
h<br />
6. A = πab<br />
<br />
a3p = 2π h2 7. Hastighed i perihelion (perigæum) : v 2 p = GM<br />
a<br />
8. Hastighed i aphelion (apogæum) : v 2 a = GM<br />
a<br />
9. va = rp<br />
ra vp<br />
1<br />
r2 = − 4π2C r2 1+e<br />
1−e<br />
1−e<br />
1+e<br />
= GMm<br />
2E<br />
10. Energi : E = Eo + 1<br />
2 (α2 − 1)mv 2 o = Eo(2 − α 2 ) (< 0 for ellipse)<br />
11. Eo = 1<br />
2 mv2 o − GMm<br />
ro<br />
12. α = vp<br />
vo<br />
13. vo =<br />
GM<br />
ro<br />
14. a = − GMm<br />
2E<br />
15. 1<br />
r = Gm2 M<br />
L 2<br />
16. 1<br />
r<br />
1 = p (1 + e cos θ)<br />
17. p = L2<br />
18. e =<br />
Gm 2 M<br />
(Energi for cirkelbevægelse)<br />
(hastighed i cirkelbevælse)<br />
<br />
1 + 1 + 2EL2<br />
G2m3M 2<br />
<br />
1/2<br />
cos θ<br />
<br />
1 + 2EL2<br />
G 2 m 3 M 2<br />
19. Impulsmoment L = mr 2 ˙ θ = mr 2 ω er konstant<br />
20. E = 1<br />
2mv2 − GMm<br />
r<br />
1. Fjeder: U(x) = 1<br />
2 kx2<br />
2. Homogen kugle:<br />
er konstant<br />
U(x) = −G Mm<br />
x<br />
U(x) = −G Mm<br />
<br />
3 −<br />
2R<br />
x2<br />
R2 <br />
3. Konstant gravitationsfelt: U(h) = mgh<br />
for x > R ⇒ F = −G(Mm/x 2 )ex<br />
for x < R ⇒ F(x) = −G Mm<br />
x<br />
R3
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