complex Circular Motion: in this problem you will generate a parametric descript

Question

complex Circular Motion: in this problem you will generate a parametric description of the motion of a point on the surface of a rotating moon orbiting a planet, which is orbiting a star. Note: The values used here are not physically accurate! Also, we will be assuming orbits are circular which, as units with period 360 days. The moon in turn orbits its planet at a radius of 10 units and period 28 days. Finally, the moon is rotating; we will assume its radius is 1 unit and its period is 5 days. Assume that all rotations are proceeding counterclockwise.

A.)   Give the parametric description for the center of the planet orbiting its star.

b.) give the parametric description for the center of the moon orbiting its planet.

c.) give the parametric description for an equatorial point on the surface of the moon rotating about the center of the moon.

D.) the combined motion is obtained by adding these separate motion. Give the final parametric description of the point on the surface of the moon.

Explanation / Answer

For the planet:
Let r = the radius of the orbit = 100
Let = the angle measured from the positive x axis
Let x = the x coordinate of the planet = (r)cos()
Let y = the y coordinate of the planet = (r)sin()
At t = 360 days, = 2
= 2t/360
a) The parametric equations of the planet where t is time (in days) is:
x = (100) cos(2t/360)
y = (100) sin(2t/360)
b) The parametric description for the center of the moon orbiting its planet

Let u = the x coordinate relative to the planet = (10) cos(2t/28)
Let v = the y coordinate relative to the planet = (10) sin(2t/28)

c) In relation to the sun, the parametric equations are the sum of the two:
x = (10) cos(2t/28) + (100) cos(2t/360)
y = (10) sin(2t/28) + (100) sin(2t/360)

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