For a circular orbit, derive Kepler's third law from scratch, using F= ma
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g., the Sun) and \( m \) (mass of the orbiting body, e.g., a planet) separated by a distance \( r \) is given by: \[ F = \frac{GMm}{r^2} \] where \( G \) is the gravitational constant. Show more…
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Circular orbit For a circular orbit, derive Kepler's third law from scratch, using F = ma.
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Show by applying Newton's law of gravitation and the second law of mechanics that a satellite (or planet) in a circular orbit of radius $R$ around the earth (or the sun) has a period (i.e. time to complete one revolution) given by $$T^{2}=\frac{4 \pi^{2} R^{3}}{G M}$$ where $M$ is the mass of the attracting body (earth or sun). This is Kepler's third law.
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(a) Use Newton's Universal Law of Gravitation and what you know about centripetal acceleration/force to derive Kepler's Third Law for a planet in a circular orbit about the sun: T² = (4π²/GM) r³ where T is the orbital period of the planet, r is the radius of the planet's orbit, M is the mass of the sun, and G is the universal gravitational constant. (b) The earth orbits the sun once per year (365 days) and its average orbital radius is 1.50 x 10¹¹ m. Use this information and the result of part (a) to estimate the mass of the sun in kilograms.
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