Gravitational field strength (g)

Gravitational field strength, denoted as 'g', is the force per unit mass experienced by a small test mass placed at a specific point. The formula for calculating g is:

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

Where:

• g = gravitational field strength (N kg^-1)
• F = gravitational force (N)
• m = test mass (kg)

On Earth, g typically has a value of approximately 9.81 N kg^-1, also referred to as the "acceleration due to gravity."

Gravitational field strength in a radial field

In radial gravitational fields, like those around spherical objects or point masses, g varies with the distance (r) from the mass. The equation is:

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

Where:

• g = gravitational field strength (N kg^-1)
• G = gravitational constant (6.67 x 10^-11 N m² kg^-2)
• M = mass of object (kg)
• r = distance from the object's centre of mass (m)

This formula shows that g decreases as the distance r from the mass centre increases, following an inverse square law.

Worked example - Calculating the gravitational field strength of Earth.

Calculate the gravitational field strength of Earth given that the mass of Earth is 5.97 x 10²⁴ kg.

The radius of Earth is 6.37 x 10⁶ m.

Step 1: Formula

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

Step 2: Substitution and correct evaluation

$\text{g = }\frac{6.67\times10^{-11}\times5.97\times10^{24}}{6.37\times10^6}\text{ = 9.81 N kg}^{-1}$

Worked example - Estimating the mass of Mars

Calculate the mass of mars given that the gravitational field strength at the surface is 3.7 N kg^-1.

The radius of Mars is 3.4 x 10⁶ m.

Step 1: Rearranged formula

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

Step 2: Substitution and correct evaluation

$M = \frac{3.7 \times (3.4 \times 10^6)^2}{6.67 \times 10^{-11}}= 6.41\times10^{23}\text{ kg}$