lecture02 - Conservation of ang mom = Kepler’s 2nd law...

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Unformatted text preview: Conservation of ang. mom. = Kepler’s 2nd law Consider a satellite’s motion over a short pe- riod of time δt . The center of the earth and its positions at the two times form a triangle l θ Postition vector at initial time Position vector time δt later r δ r The area of the triangle is given by δA = rl/ 2 = 1 2 rδr sin θ = 1 2 | r × δ r | . The rate that area is swept out is ˙ A = lim δt → δA δt = lim δt → 1 2 | r × δ r | δt = 1 2 | r × ˙ r | = 1 2 h. The area swept out over a finite time interval t 1 to t 2 is the integral A t 1 ,t 2 = Z t 2 t 1 1 2 hdt = 1 2 h ( t 2- t 1 ) is always the same (depends only on t 2- t 1 ); This is Kepler’s second law. L. Healy – ENAE404 – Spring 2007 – Lecture 2 (Jan. 30) 1 Planar motion Another important consequence of the conser- vation of angular momentum is that the orbit stays in a single plane because h ⊥ r and h ⊥ ˙ r , so both the change of position and position lie in the plane perpendicular to the angular momentum, which is constant. This plane of motion is defined by the perpendicular to the angular momentum vector. Since the velocity vector lies in this plane, the position vector can never move out of it. This is implied in Kepler’s first law — an el- lipse is a planar figure — but he didn’t state it explicitly. L. Healy – ENAE404 – Spring 2007 – Lecture 2 (Jan. 30) 2 Angular speed and angular momentum Since angular momentum is the cross product of position and velocity vectors, it can be ex- pressed as the product of magnitudes and the sine of the angle between them θ , h = rv sin θ. The time rate of change of the satellite’s angu- lar position times the distance gives the com- ponent of velocity perpendicular to the posi- tion vector, r ˙ θ = v sin θ, matching the component of linear speed in the direction perpendicular to the position vector. Substituting for the right side in the first equa- tion, h = r 2 ˙ θ (Curtis (2–37)). This relationship will be used later....
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lecture02 - Conservation of ang mom = Kepler’s 2nd law...

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