4.1 Gravity  (Page 31/6)

The term “gravity” is used for the gravitation between two bodies, one of which is Earth.

Earth is composed of layers, having different densities and as such is not uniform. Its density varies from 2 $kg/m^3$ for crust to nearly 14 $kg/m^3$ for the inner core. However, inner differentiation with respect to mass is radial and not directional. This means that there is no preferential direction in which mass is aggregated more than other regions. Applying Newton’s shell theorem, we can see that Earth, if considered as a solid sphere, should behave as a point mass for any point on its surface or above it.

In the nutshell, we can conclude that density difference is not relevant for a point on the surface or above it so long Earth can be considered spherical and density variation is radial and not directional. As this is approximately the case, we can treat Earth, equivalently as a sphere of uniform mass distribution, having an equivalent uniform (constant) density. Thus, force of gravitation on a particle on the surface of Earth is given by :

$$F=\frac{GMm}{R^2}$$

where “M” and “m” represents masses of Earth and particle respectively. For any consideration on Earth’s surface, the linear distance between Earth and particle is constant and is equal to the radius of Earth (R).

Gravitational acceleration (acceleration due to gravity)

In accordance with Newton’s second law of motion, gravity produces acceleration in the particle, which is situated on the surface. The acceleration of a particle mass “m’, on the surface of Earth is obtained as :

$$\Rightarrow a=\frac{F}{m}=\frac{GM}{R^2}$$

The value corresponding to above expression constitutes the reference gravitational acceleration. However, the calculation of gravitational acceleration based on this formula would be idealized. The measured value of gravitational acceleration on the surface is different. The measured value of acceleration incorporates the effects of factors that we have overlooked in this theoretical derivation of gravitational acceleration on Earth.

We generally distinguish gravitational acceleration as calculated by above formula as “ $g_0$ ” to differentiate it from the one, which is actually measured(g) on the surface of Earth. Hence,

$$g_0=a=\frac{F}{m}=\frac{GM}{R^2}$$

This is a very significant and quite remarkable relationship. The gravitational acceleration does not dependent on the mass of the body on which force is acting! This is a special characteristic of gravitational force. For all other forces, acceleration depends on the mass of the body on which force is acting. We can easily see the reason. The mass of the body appears in both Newton's law of motion and Newton's law of gravitation. Hence, they cancel out, when two equations are equated.

Factors affecting gravitational acceleration

The formulation for gravitational acceleration considers Earth as (i) uniform (ii) spherical and (iii) stationary body. None of these assumptions is true. As such, measured value of acceleration (g) is different to gravitational acceleration, “ $g_0$ ”, on these counts :