Between Mars' and Jupiter's orbits, the asteroid belt revolves around the sun. A single asteroid...

Gregory Emery

Gregory Emery

Answered

2022-01-07

Between Mars' and Jupiter's orbits, the asteroid belt revolves around the sun. A single asteroid orbits the earth every 5.0 years.
What are the orbital radii and velocities of the asteroids?

Answer & Explanation

Jimmy Macias

Jimmy Macias

Expert

2022-01-08Added 30 answers

Given values:
T1=5years=5365243600s=1.5768108s
G=6.671011Nkg2kg2
msun=1.991030kg
The fravitational force between the sun and the orbiting asteroid is:
F=Gmsunmar12
The asteroid is affected by centripetal acceleration because it moves in a circular orbit
F=mac,
ac=v2r
According to the Newtons
trisanualb6

trisanualb6

Expert

2022-01-09Added 32 answers

Using Kepler's 3rd law which is: T2=4π2r3GM
Solved for r:
r=[GMT24π2]13
Where G is the universal gravitational constantM is the mass of the sun, T is the asteroid's period in seconds, and r is the radius of the orbit.
Change 5.00 years to seconds :
5.00 years=5.00 years (365days/year)(24.0hours/day)(6=1.58×108s
The radius of the orbit then is computed:
r=[(6.67×1011Nmkg2)(1.99×1030kg)(1.58×108s)24π2]13=4.38×1011m

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