Show how the Law of Universal Gravitation can be derived from Kepler’s Third Law and Newton’s First Law
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Mathematically, this can be expressed as: \[ T^2 \propto a^3 \] or \[ T^2 = k \cdot a^3 \] where \( k \) is a constant that depends on the mass of the central body (e.g., the Sun). Show more…
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Kepler's Laws, impressive as they are, were purely descriptive. Newton's great achievement was to find an underlying cause for them. In this project, you will derive Kepler's Second Law from Newton's Law of Gravity. Consider a coordinate system centered at the sun. Let r be the position vector of the planet, and let v and a be the planet's velocity and acceleration, respectively. Define L as the planet's angular momentum. (a) Show that dL/dt = r x a. (b) Consider the planet moving from r to r + dr. Explain why the area dA about the origin swept out by the planet approximately equals 1/2 ||dr x r||. (c) Using the result from (b), explain why dA/dt = 1/2 ||L||. (d) Newton's Laws imply that the planet's gravitational acceleration, a, is directed toward the sun. Using this fact and the result from (a), explain why L is constant. (e) Use the results from (c) and (d) to explain Kepler's Second Law. (f) Using Kepler's Second Law, determine whether a planet is moving most quickly when it is closest to, or farthest from, the sun. Figure 17.43: The line segment joining a planet to the sun sweeps out equal areas in equal times
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