Revise Gravitational field for Cambridge 9702 (13.1)

A gravitational field is a region of space where any object possessing mass will experience an attractive gravitational force due to another mass. These fields are inherently created by masses, and they're always attractive, never repulsive. To visualise this invisible influence, we use gravitational field lines. These lines have specific properties: they point in the direction of the force on a test mass, their density (how close they are) represents the field's strength, and they never cross. For a single point mass, the field lines are radial, pointing inwards.

Gravitational field strength, denoted by 'g', is defined as the gravitational force experienced per unit mass at a particular point within the field. Its SI units are Newtons per kilogram (N kg⁻¹), which is equivalent to metres per second squared (m s⁻²). This equivalence highlights that 'g' also represents the acceleration a body would undergo if it were to fall freely within that field. Near Earth's surface, 'g' is approximately uniform, but further away from large masses, fields become radial, with strength varying significantly with distance.

$g = \frac{F}{m}$

Sir Isaac Newton formulated the universal law describing the attractive force between any two point masses. This law states that the gravitational force (F) between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance separating their centres. This groundbreaking law allows us to calculate the force between any two objects in the universe.

$F = G \frac{m_1 m_2}{r^2}$

For a radial field, like the one created by a planet, we can combine the definitions of 'g' and Newton's Law of Gravitation. If 'M' is the mass creating the field and 'm' is a small test mass, then $F = GMm/r^2$. Since $g = F/m$, we can derive a specific formula for the gravitational field strength in a radial field. This clearly shows that 'g' also follows an inverse square law, meaning its strength diminishes significantly with increasing distance from the central mass.

$g = \frac{GM}{r^2}$

Gravitational potential ($\phi$) at a point is defined as the work done per unit mass required to bring a unit mass from infinity to that specific point within the field. Conventionally, the gravitational potential at an infinite distance from a mass is set to zero. As energy is released (work is done by the field) when a mass moves from infinity into the field, gravitational potential is always a negative value. Think of it as 'energy debt' – the deeper into the field, the more negative the potential.

$\phi = -\frac{GM}{r}$

Gravitational potential energy ($E_p$) is the total energy an object possesses due to its position within a gravitational field. It's the work done to move a mass from infinity to that point. This means it's simply the gravitational potential ($\phi$) multiplied by the mass of the object ('m'). Like potential, it's also a negative value, representing the binding energy of the system. For orbiting objects, the gravitational force acts as the centripetal force.

$E_p = -\frac{GMm}{r}$

Understanding how gravitational field strength (g) and gravitational potential ($\phi$) vary with distance (r) from a central mass M is crucial. Their graphical representations reveal key aspects of the inverse square law. For a spherical mass, we consider the distance r from its centre.