. Consider a planet with mass Mptanet to orbit in nearly circular motion about the sun of mass Msun: Prove that T² = * using the following equations and GMsun conditions on orbital motion Net centripetal force: Fnet Mptanet v² R GMplanetMsun Gravitational force: Fgrav = Fgrav = Fnet Obtain the equation of speed along a circular path, v. Substitute the equation of speed, v to the period of revolution: T 2mR Simplify the equation, square both sides and it should yield T² 4x²R %3D GMsun

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Orbits and Planetary Motion. Directions: Answer the following problem involving
orbits and planetary motion. Refer to information given in the box below. Show your
solution and box your final answer
|Kepler's three laws of planetary motion can be described as follows:
• The path of the planets about the sun is elliptical in shape, with the center of
the sun being located at one focus. (The Law of Ellipses)
An imaginary line drawn from the center of the sun to the center of the planet
will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods (T²) of any two planets is equal to the
ratio of the cubes of their average distances from the sun (R³). (The Law of
Harmonies):
3. Consider a planet with mass Mplanet to orbit in nearly circular motion about the
sun of mass Msun. Prove that T² = 4"*R²
using the following equations and
GMsun
conditions on orbital motion
Mptanet v²
• Net centripetal force: Fnet =
Gravitational force: Fgrav
GMplanetMsun
R2
Fgrav = Fnet
Obtain the equation of speed along a circular path, v.
Substitute the equation of speed, v to the period of revolution: T = 2
Simplify the equation, square both sides and it should yield T :
GMsun
47²R³
Show your derivation to prove that T²
%3D
GM sun
Transcribed Image Text:Orbits and Planetary Motion. Directions: Answer the following problem involving orbits and planetary motion. Refer to information given in the box below. Show your solution and box your final answer |Kepler's three laws of planetary motion can be described as follows: • The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods (T²) of any two planets is equal to the ratio of the cubes of their average distances from the sun (R³). (The Law of Harmonies): 3. Consider a planet with mass Mplanet to orbit in nearly circular motion about the sun of mass Msun. Prove that T² = 4"*R² using the following equations and GMsun conditions on orbital motion Mptanet v² • Net centripetal force: Fnet = Gravitational force: Fgrav GMplanetMsun R2 Fgrav = Fnet Obtain the equation of speed along a circular path, v. Substitute the equation of speed, v to the period of revolution: T = 2 Simplify the equation, square both sides and it should yield T : GMsun 47²R³ Show your derivation to prove that T² %3D GM sun
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