Chapter 5 | Integration
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202. Kepler’s first law states that the planets move in elliptical orbits with the Sun at one focus. The closest point of a planetary orbit to the Sun is called the perihelion (for Earth, it currently occurs around January 3) and the farthest point is called the aphelion (for Earth, it currently occurs around July 4). Kepler’s second law states that planets sweep out equal areas of their elliptical orbits in equal times. Thus, the two arcs indicated in the following figure are swept out in equal times. At what time of year is Earth moving fastest in its orbit? When is it moving slowest?
206. The displacement from rest of a mass attached to a spring satisfies the simple harmonic motion equation x ( t ) = A cos ⎛ ⎝ ωt − ϕ ⎞ ⎠ , where ϕ is a phase constant, ω is the angular frequency, and A is the amplitude. Find the average velocity, the average speed (magnitude of velocity), the average displacement, and the average distance from rest (magnitude of displacement) of the mass.
203. A point on an ellipse with major axis length 2 a and minor axis length 2 b has the coordinates ( a cos θ , b sin θ ), 0≤ θ ≤2 π . a. Show that the distance from this point to the focus at (− c , 0) is d ( θ ) = a + c cos θ , where c = a 2 − b 2 . b. Use these coordinates to show that the average distance d – from a point on the ellipse to the focus at (− c , 0), with respect to angle θ , is a . 204. As implied earlier, according to Kepler’s laws, Earth’s orbit is an ellipse with the Sun at one focus. The perihelion for Earth’s orbit around the Sun is 147,098,290 km and the aphelion is 152,098,232 km. a. By placing the major axis along the x -axis, find the average distance from Earth to the Sun. b. The classic definition of an astronomical unit (AU) is the distance from Earth to the Sun, and its value was computed as the average of the perihelion and aphelion distances. Is this definition justified? 205. The force of gravitational attraction between the Sun and a planet is F ( θ ) = GmM r 2 ( θ ) , where m is the mass of the planet, M is the mass of the Sun, G is a universal constant, and r ( θ ) is the distance between the Sun and the planet when the planet is at an angle θ with the major axis of its orbit. Assuming that M , m , and the ellipse parameters a and b (half-lengths of the major and minor axes) are given, set up—but do not evaluate—an integral that expresses in terms of G , m , M , a , b the average gravitational force between the Sun and the planet.
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