Acceleration Due to Gravity Below and Above the Surface of Earth
This section elaborates on the concept of gravitational acceleration (g) at different positions relative to Earth’s surface. It separates the discussion into two main areas: acceleration at heights above the Earth's surface and acceleration at depths below the surface.
Gravitational Force Above Earth's Surface
When a mass m is positioned at a height h above the Earth, the gravitational force acting on it can be described by the formula:
\[ F(h) = \frac{GM_E m}{(R_E + h)^2} \]
where G is the gravitational constant and \( R_E \) is the radius of the Earth. From this force, we derive acceleration due to gravity:
\[ g(h) = \frac{F(h)}{m} = \frac{GM_E}{(R_E + h)^2} \]
This indicates that as h increases, g(h) decreases relative to the surface gravity, diminishing with height.
For small heights (\( h << R_E \)), we can approximate g(h) using a binomial expansion:
\[ g(h) \approx g - \frac{g h}{R_E} \]
where g is the gravitational acceleration at the Earth's surface.
Gravitational Force Below Earth's Surface
Conversely, at a depth d below the surface, the calculation changes. The force on a mass m at that depth can be expressed as:
\[ F(d) = \frac{GM_S m}{(R_E - d)^2} \]
where \( M_S \) is the mass of the smaller sphere of radius \( (R_E - d) \) that contributes to the gravitational pull. The total gravitational acceleration then becomes:
\[ g(d) = g \left(1 - \frac{d}{R_E}\right) \]
This depicts how gravitational acceleration diminishes linearly as one travels deeper into the Earth. Notably, gravity is maximal at the surface and decreases consistently as one moves both upward to heights and downward into depths. Thus, the behavior of gravitational acceleration is fundamental for understanding various physical phenomena related to gravity.