Gravitational Field Intensity

Gravitational field strength is the gravitational force per unit mass at a particular point. Gravitational field strength 

G =\( \frac{Gravitational \; force}{mass} = \frac{F}{m}\)

F = mg

The gravitational field strength (g) is also known as acceleration due to gravity. The unit of g is N/kg and is constant at a particular location.

Variation of g with Distance (r)

The attractive force of the earth on another mass (m) is related to distance (r) from the earth by:

 F = \( \frac{GMM} {r^2}\) ________ (1)

 F=mg _______ (2)

Combining (1) and (2)

 Mg = \( \frac{GMM} {r^2}\)   ___________  (3)

 g = \( \frac{GM} {r^2}\) __________ (4)

This equation four shows that 

• g is inversely proportional to the square of their distance apart
• Varies as the radius of the earth
• Obeys inverse square law as g decreases as mass distance from the earth increases

Variation of g with Height

As the mass moves away from the center of the earth, g also decreases

 I.e. g = \( \frac{GM} {r^2}\) _________  (1)

At a height h from the earth surface, 

 g=\( \frac{GM}{r + h^2} = \frac{GM}{R^2}\)  (2)

Dividing (1) by (2)

\( \frac{g^1}{g} = \frac{GM}{(r + h^2)} = \frac{r^2}{R^2}\)  (2)

\( \scriptsize g^1 = g \left(\normalsize \frac{r}{R} \right)^2 \)

Example

The acceleration due to gravity near the earth surface is about10ms^-2. Calculate the gravitational field strength at a height (h) twice the radius of the earth.

\( \scriptsize g^1 = g \left(\normalsize \frac{r}{R} \right)^2 \)

\( \scriptsize g^1 = 10 \left(\normalsize \frac{r}{R} \right)^2 \)

\( \scriptsize g^1 = 10 \left(\normalsize \frac{1}{2} \right)^2 \)

= 2.5ms^-2