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Understanding the universal law of gravitation is pivotal to solving problems related to celestial motion. Initially formulated by Sir Isaac Newton, it states that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

This law can be represented by the formula:
\[F_{grav} = \frac{G \times M_1 \times M_2}{r^2}\]
where:\

• \(F_{grav}\) is the gravitational force between the two masses,
• \(G\) is the gravitational constant (\(6.674 \times 10^{-11} N(m/kg)^2\)),
• \(M_1\) and \(M_2\) are the masses of the two objects, and
• \(r\) is the distance between the centers of the two masses.

When approaching problems involving the motion of a spacecraft, such as Apollo 8, around a celestial body like the Moon, this law allows us to calculate the gravitational pull exerted by the Moon on the spacecraft, which is essential for determining the orbit characteristics.

The term 'centripetal' comes from Latin words that mean 'seeking the center'. Centripetal force is required for an object to move in a circular path and is directed towards the center around which the object is moving. This force is not a separate force of nature but rather the net force causing the circular motion. For a spacecraft in orbit, like Apollo 8, this force is provided by the gravitational pull of the celestial body it orbits.

The relationship between centripetal force \(F_{centripetal}\) and the velocity of an object is given by the equation:
\[F_{centripetal} = m \times \frac{v^2}{R_total}\]
where:\

• \(m\) is the mass of the orbiting object,
• \(v\) is its velocity, and
• \(R_{total}\) is the radius of the circular orbit.

In the case of the Apollo 8 mission, we can use the equivalence of gravitational and centripetal forces to find the required velocity and hence, with the orbital radius, calculate the orbital period.

The calculation of gravitational force comes into play when determining the velocity necessary for a spacecraft to maintain orbit around a celestial body like the Moon. The steps in the provided solution employ gravitational force calculation to reach an understanding of orbital mechanics.

By equating the gravitational force \(F_{grav}\), which acts as the centripetal force keeping Apollo 8 in orbit, with the centripetal force formula, we eliminate the need for the spacecraft's mass in calculating its velocity, which simplifies the problem immensely:
\[\frac{G \times M_{moon}}{R_{total}} = \frac{v^2}{R_{total}}\]
After finding the velocity, the period \(T\) for one complete orbit can be found using the relationship:
\[T = \frac{2 \pi R_{total}}{v}\]
This process demonstrates the practical application of gravitational force calculation in predicting and analyzing the motion of satellites and spacecrafts in orbit.