, an inverse square law.

Transcription

1 Uniform irular motion Speed onstant, but eloity hanging. and a / t point to enter. s r θ > θ s/r t / r, also θ in small limit > t/r > a / r, entripetal aeleration Sine a points to enter of irle, F m a - m / r rˆ, where rˆ is a unit etor in the radial diretion. Now, sine π r / P, and P k r 3 Newton realized that m F 4 π gra kr, an inerse square law.

2 Newton s Law of Graitation Mutual attratie fore between any two point masses m and m along the line joining them, Gm m Fgra. r It turns out that G 6.67 x 0 - m 3 /(kg s ). A key extension of this result: The graitational fore exerted by a spherial body is the same as if it s entire mass were onentrated at it s enter. Therefore, at the Earth s surfae, F gra GM E m R E mg, where g GM E /R E 9.80 m/s

3 Kepler s Laws and Newtonian Mehanis () Elliptial orbits F m a and F GMm/r > orbits are oni setions. Kepler s st Law a speial ase. () Law of Areas θ t t/r for small t Also, A / r r θ / r t t da/dt / r t. This quantity is onstant aording to Kepler s nd Law. Why?

4 Kepler s Laws and Newtonian Mehanis Newtonian deriation of nd Law: Angular momentum L r x p m (r x ) Cross produt: magnitude r p sin θ, where θ is relatie angle of r, p diretion by right hand rule Torque dl/dt x p + r x dp/dt r x F 0 if F is radial. Therefore, L m r sin θ m r t m H onstant, where H r t. da dt r t H onstant, proing nd Law. Can also write H da dt A P πab. P

5 Consequenes: At perihelion and aphelion, the eloity is transerse to radius etor. H r t r p p r a a > p a πab π ab P rp P a( e), πab πab P ra P a( + e). Use b a(-e ) / > π a + p P a πa P + e e e e / /,.

6 (3) Newton s form of Kepler s 3rd Law Both objets orbit around CM with period P. Combine F m /r, F m /r,, r mr Two objets orbit around their stationary enter of mass (CM) at distanes r and r. CM stays fixed (or moes at onstant eloity) if no external fores P π r / π r /, and F F to obtain m whih defines position of CM.

7 More omplete form of Kepler s 3rd Law - takes into aount motion of both bodies. Now a r + r. m m m a r + Also, Newton s Law of Graity yields a Gm m F F F gra. ) ( m m G a P F m r P + π π

8 Using Kepler s 3rd Law Consider two objets in two separate orbits. Form ratios of preious equation: ( ) [ ' ' ]( ) P / P' ( m + m ) / ( m + m ) a / a'. For two objets orbiting the Sun, 3 ' m + M m + M M Sun Sun Sun, so that ( P / P') ( a / a') 3, where P yr and a AU. e.g., Mars, a.5 AU > P Mars (.5) 3/ yr.88 yr.

9 Orbital Veloity Preious expressions for H r t yield r d θ π a t θ dt P r r a( e ) + eosθ ( e ) / Combine with polar equation for ellipse to obtain dr πa ( ) dt P e e / r sinθ r d a e θ π θ ( + osθ ) dt P ( e ) /

10 The total orbital speed is then a e r + π + osθ + θ P e e Eliminate os θ using the polar equation of an ellipse to obtain G ( m + m ) r a The is ia equation. This equation is atually a statement of energy onseration.

11 Conseration of Energy in a Two-Body System Orbital speed used preiously is the relatie speed of the two bodies. +, where and are the indiidual speeds. Also, m m (Newton s st Law) whih leads to ( m + m ) ( m + m ). m m Substituting into the isia equation yields m Gm m Gmm + m. r a st two terms: Kineti Energy (KE) of two objets around CM 3rd term: Potential Energy (PE) of the interation RHS: The total energy. It is onsered for an isolated system. Also, negatie for a bound system.

12 Satellite orbits At the Earth s surfae, get esape speed from energy onseration: ( m ) ( GM m R ) 0, i.e., the ase where drops to 0 at an infinite radius > ( GM R ) es The irular speed at the Earth s surfae is ( ) / GM R, so that. 0 es 0 In general, a satellite reahes its effetie launh eloity far aboe the Earth s surfae, when its fuel runs out. The subsequent orbit depends on the relation of its speed to the irular speed and esape speed at that radius. /.

13 Satellite orbits GM r r R +, where h, h is the height aboe surfae. Let ( ) / Then es at this radius. If the burnout eloity is parallel to the Earth s surfae, we get the following possible outomes by studying the is ia equation, ( ) r a a < < / a, parabola < > r ( ) elliptial orbit, start at apogee irular orbit elliptial orbit, start at perigee hyperbola :