To understand the relationship between the forces [tex]\(F_{P1}\)[/tex] and [tex]\(F_{P2}\)[/tex] acting on two planets in the same circular orbit around a central star, we will use the universal law of gravitation.
According to the universal law of gravitation, the force [tex]\(F\)[/tex] between any two masses [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] is given by the formula:
[tex]\[
F = G \cdot \frac{m_1 \cdot m_2}{r^2}
\][/tex]
where:
- [tex]\(G\)[/tex] is the gravitational constant,
- [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are the masses of the two bodies,
- [tex]\(r\)[/tex] is the distance between the centers of the two bodies.
Given:
1. The mass of planet 1 ([tex]\(m_1\)[/tex]) is equal to the mass of planet 2 ([tex]\(m_2\)[/tex]).
2. Both planets are in the same circular orbit around the central star, meaning the distance [tex]\(r\)[/tex] between the planets and the star is the same.
Since the distance and the masses are the same, the gravitational force between the star and each planet depends only on the universal gravitation formula. Therefore, the force exerted by the star on planet 1 ([tex]\(F_{P1}\)[/tex]) will be exactly the same as the force exerted by the star on planet 2 ([tex]\(F_{P2}\)[/tex]).
Thus, the relationship between [tex]\(F_{P1}\)[/tex] and [tex]\(F_{P2}\)[/tex] is:
[tex]\[
F_{P1} = F_{P2}
\][/tex]
Therefore, the correct answer is:
(B) [tex]\(F_{P1} = F_{P2}\)[/tex].
