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Understanding Newton's law of universal gravitation is central to grasping the fundamental forces that govern celestial bodies. This law posits that every mass exerts an attractive force on every other mass. The strength of this gravitational pull is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers.

This concept is neatly captured by the formula: \[ F = G \frac{m_1 m_2}{r^2} \] Here, \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses. When studying the gravitational attraction between two objects, this law allows us to predict the force with which they will draw each other closer.

The gravitational constant, symbolized as \( G \), is a key factor in calculating gravitational forces. It is a universal value that determines the strength of gravity in the equation of Newton's law of universal gravitation.

The gravitational constant has been experimentally measured to be approximately \( 6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \). Although this number seems very small, it's crucial to understand that the effects of gravity become significant over large masses or over short distances. Whenever we use the formula for gravitational force, the constant \( G \) helps us to ensure that we get the force value in newtons, providing consistency and reliability in calculations across all gravitational problems.

The relationship between mass and radius in the context of gravitational force is vital to comprehend. This relation comes into play when we're dealing with spherical objects, where the mass distribution is assumed to be uniform.

An important point to note is that the gravitational force does not depend on the surface area of the spheres but rather on the radius—or distance to the center—where all the mass can be assumed to be concentrated for the purpose of these calculations. This principle dramatically simplifies the calculations for gravitational force between spherical bodies, because we can treat them as point masses located at their centers, as long as we are only concerned with the gravitational forces they exert on each other.

When it comes to calculating the gravitational force between two uniform spheres, such as in our given problem, it is important to apply Newton's law with the understanding that the distance used in the computation is the distance between the spheres' centers.

In our exercise, since the spheres are touching, the distance \( r \) is equal to twice the radius \( R \), or \( 2R \). Subsequently, because the spheres are uniform and identical with mass \( M \), our formula simplifies to \[ F = G \frac{M^2}{(2R)^2} = G \frac{M^2}{4R^2} \]. This simplified expression underscores the inverse square relation of the force to the distance between the centers of the two spheres. The calculation demonstrates how an increase in distance would decrease the gravitational force, illuminating the delicate balance of gravitational interactions in our universe.