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The universal gravitation formula is a critical concept to understand when delving into the forces that govern astronomical bodies and objects on Earth. Sir Isaac Newton articulated this fundamental principle in his law of universal gravitation, which posits that every point mass attracts every other point 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.

In mathematical terms, this can be represented as:

\[F = G \frac{{m_1 \times m_2}}{{d^2}}\]

where:

• \(F\) is the force of gravity between two objects,
• \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \, m^{3} \, kg^{-1} \, s^{-2}\)),
• \(m_1\) and \(m_2\) are the masses of the two objects,
• \(d\) is the distance between the centers of the two masses.

Understanding this equation is crucial for calculations involving the gravitational attraction between any two bodies, such as a person and the Earth, or—as in our exercise—a person and a massive water tank.

Calculating the gravitational force between two bodies involves applying the universal gravitation formula mentioned earlier. It's a straightforward calculation once values for mass and distance are known or can be estimated. If we consider the person standing below the water tank in our exercise, the masses (human and water) and the distance between them form the base for our calculations.

Plugging the known values into the equation:

\[F = G \frac{{m_1 \times m_2}}{{d^2}} = 6.674 \times 10^{-11} \frac{{(4 \times 10^{6})(70)}}{{(15)^2}}\]

will yield the gravitational force exerted by the water tank on the person. In this context, it's important to remember that whilst the gravitational force might seem minuscule for small masses or at great distances, it becomes substantially significant for larger masses or when bodies are in close proximity, justifying its key role in astronomy and on Earth's surface.

When we speak of weight reduction calculation due to gravitational forces other than Earth's, we are referring to the idea that a nearby massive object can exert a force on us, thereby slightly altering the net gravitational force and, consequently, our apparent weight. In the exercise, we measure this effect of the water's gravitational pull as a fraction of the person's overall weight. Since weight is the force due to gravity, calculated as:

\[mg\]

where \(m\) is mass and \(g\) is the acceleration due to gravity, the fractional weight reduction due to another object's gravitational force is:

\[\frac{F}{mg}\]

Here, \(F\) denotes the gravitational force between the person and the water tank, while \(mg\) represents the person's weight due to Earth's gravity. This ratio provides the fractional change in the person's weight resulting from the gravitational attraction to the water tank.