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Understanding the concept of uniform spheres is essential when discussing gravitational force. A uniform sphere is an object that has a symmetrical shape with all its mass evenly distributed throughout its volume. This means that the center of mass is located precisely at the geometric center of the sphere. This property simplifies our calculations related to gravitational interactions because we can treat the entire sphere as if all its mass were concentrated at its center.
In the context of gravitational force, assuming uniformity in spheres allows us to use the gravitational force formula effectively. Also, it simplifies our understanding of how gravitational attraction occurs between spherical objects, like planets or smaller spheres, in a void of space.
Uniform spheres are a fundamental assumption in many physics problems because they allow us to replace complex real-world shapes with simple calculations.

The gravitational constant, denoted as \( G \), is a crucial factor in calculating gravitational forces. It is a constant value that applies universally, making it integral for understanding gravitational attraction between masses.
The value of \( G \) is approximately \( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \), and it represents the strength of gravity in classical physics. The presence of \( G \) in the gravitational force formula indicates that regardless of the individual properties of the objects (like mass or distance), gravity's effects are predictable and consistent.
When tackling problems involving gravitational force, \( G \) ensures that we can account for the cosmic-scale attraction phenomena between celestial bodies, as well as small-scale interactions. This consistency is a cornerstone of Newtonian physics.

The gravitational force formula is the backbone of many calculations related to gravitational attraction. Mathematically, it is expressed as \( F = \frac{G m_1 m_2}{r^2} \), where:\[ F \] is the gravitational force between two masses,
\( G \) is the gravitational constant,
\( m_1 \) and \( m_2 \) are the masses of the interacting bodies,
\( r \) is the distance between the centers of these masses.
The formula indicates that gravitational force increases with greater mass, but decreases as the distance between the two objects increases. This is because the force is inversely proportional to the square of the distance, meaning a larger separation results in a weaker force.
By substituting known values into this formula, we can determine the gravitational force in various contexts, from touching spheres to distant celestial bodies.

The distance between the centers of mass of two objects plays a pivotal role in calculating gravitational force. For uniform spheres touching each other, this distance is simply the sum of their radii, as they span from one center to another directly through their touching surfaces.
This factor of distance, represented as \( r \), in the gravitational force formula \( F = \frac{G m_1 m_2}{r^2} \), shows that even slight increases in separation can significantly decrease the force of attraction. It is because the force is inversely proportional to the square of the distance — doubling the distance would quarter the force.
In the given exercise, because the spheres have a radius \( R \), when they touch, the center-to-center distance is \( 2R \). This straightforward calculation allows easy computation when using the gravitational force formula.