Equation 7.8 relates force to the derivative of potential energy. Use this fact to differentiate Equation 8.6 for gravitational potential energy, and show that you recover Newton's law of gravitation.

Gravitational potential energy (GPE) is a form of energy an object possesses because of its position in a gravitational field. For two objects of mass M and m separated by a distance r, the GPE is given by the equation \( U = -\frac{GMm}{r} \).Here, GPE is considered negative because work is done by the gravitational force to move an object to that position against the force, meaning that we would need to input positive energy to remove the object from the field, hence the 'debt' of energy.

Imagine lifting a book from the floor to a shelf; as you elevate the book, you're increasing its GPE within the Earth's gravitational field. If released, the GPE converts to kinetic energy as the book falls. This conservation of energy ensures GPE is a fundamental concept in understanding celestial mechanics and energy transitions within gravitational fields.

Differentiation is a mathematical process used in physics to determine how a function changes as its input changes. In the context of gravitational potential energy, differentiation helps us find the force between two objects. By differentiating the function for GPE with respect to the distance r between the masses, we can find the rate at which the potential energy changes, which according to physics, is the force exerted by gravity.

Thus, the derivative \( \frac{dU}{dr} \) is used to describe the relationship between gravitational potential energy and gravitational force. When we recognize that force F can be represented as \( F = -\frac{dU}{dr} \), we assert that the force is the negative gradient of potential energy, aligning with the concept that gravitational force attempts to minimize potential energy, pulling masses together.

The universal gravitation constant, denoted by G, is a key factor in the formula for both gravitational potential energy and Newton's law of universal gravitation. This constant quantifies the strength of gravity for any two objects with mass.The value of G is approximately \( 6.674 \times 10^{-11} \) N(m/kg)^2 and is derived experimentally. G appears in the gravitational force equation \( F = \frac{GMm}{r^2} \) as the proportionality factor, showing that the gravitational force is directly proportional to the product of the two masses, and inversely proportional to the square of the distance between them. This fundamental constant is essential for making precise calculations in celestial mechanics and was crucial in determining the Earth's mass, the behavior of planets, and the motion of galaxies.