Gravitational field strength Knowledge-Based Study

Study notes

1. What is Gravitational Field Strength?

   A gravitational field is a region of space surrounding any object with mass where another mass experiences an attractive gravitational force. To quantify the strength of this interaction at any given location without depending on the mass of a secondary object placed there, we define gravitational field strength (g).

   Gravitational field strength at a point is defined as the gravitational force exerted per unit mass on a small test mass placed at that point:

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

   • $F_g$: Gravitational force acting on the test mass (newtons, N)
   • $m$: Mass of the test object (kilograms, kg)
   • $g$: Gravitational field strength ($\text{N}\cdot\text{kg}^{-1}$ or $\text{m}\cdot\text{s}^{-2}$)

   Because force is a vector quantity and mass is a scalar, gravitational field strength is a vector quantity. Its direction is always radially inward toward the center of the mass producing the field.

   Deriving Field Strength for a Point or Spherical Mass

   Newton's Law of Universal Gravitation states that the attractive force between two point masses $M$ and $m$ separated by a center-to-center distance $r$ is:

   $F_g = \frac{G M m}{r^2}$

   Substituting this expression into the definition of gravitational field strength ($g = \frac{F_g}{m}$) yields the formula for the magnitude of the gravitational field strength created by a primary mass $M$:

   $g = \frac{G M}{r^2}$

   • $G$: Universal gravitational constant ($6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2\cdot\text{kg}^{-2}$)
   • $M$: Mass creating the gravitational field (kg)
   • $r$: Distance from the center of mass $M$ to the point of interest (m)

   For any spherically symmetric body, such as a planet or star, the mass behaves gravitationally as if all of it were concentrated at its geometric center for points outside its surface.

   The Inverse-Square Relationship

   The equation $g = \frac{G M}{r^2}$ demonstrates that gravitational field strength follows an inverse-square law with respect to distance from the center of mass:

   • Doubling the distance ($2r$) reduces the field strength to one-quarter ($\frac{1}{4}g$).
   • Tripling the distance ($3r$) reduces the field strength to one-ninth ($\frac{1}{9}g$).

   Role in Circular Orbital Motion

   When an object orbits a planet in a circular path of radius $r$, the gravitational force provides the necessary centripetal force:

   $F_g = F_c \implies m g = \frac{m v^2}{r}$

   Dividing both sides by the orbiting mass $m$ reveals that the gravitational field strength at that altitude is equal to the centripetal acceleration of the satellite:

   $g = a_c = \frac{v^2}{r}$

   Consequently, the required orbital speed depends solely on the central mass $M$ and orbital radius $r$, completely independent of the satellite's own mass.