7.6. ACCELERATION DUE TO GRAVITY BELOW AND ABOVE THE SURFACE OF EARTH

Consider a point mass m at a height h above the surface of the earth as shown in Fig. 7.8(a). The radius of the earth is denoted by RE. Since this point is outside the earth,

$F(h) = \frac{G M_E m}{(R_E + h)^2}$.

The acceleration experienced by the point mass is $g(h) = \frac{F(h)}{m}$ and we get

$g(h) = \frac{G M_E}{(R_E + h)^2}$

This is clearly less than the value of g on the surface of earth: $g_E = \frac{G M_E}{R_E^2}$.

When a mass is at a height h above the Earth's surface, we can derive the gravitational force using Newton's law of gravitation. This force diminishes with distance, illustrated with the formula for gravitational force at height h: F(h) = (G M_E m) / (R_E + h)², where G is the gravitational constant and M_E is Earth's mass. To determine the acceleration experienced by the mass at this height, we rearrange this equation to isolate acceleration, leading to the formula g(h) = (G M_E) / (R_E + h)². This shows that as we move further away from Earth's surface, the gravitational pull (and hence acceleration) decreases compared to the value at the surface.

Imagine being on a high mountain or in a plane. The higher you go, you feel lighter. This is similar to how the pull of Earth's gravity decreases with height, similar to how the force of a magnet decreases as you move it further away.

For small heights h where $h<<R_E$, we can expand the RHS of the above equation:

$g(h) \approx g_E \left(1 - \frac{h}{R_E}\right)$.

This chunk introduces a more refined approximation for gravitational acceleration at minor heights above the Earth's surface. By using a binomial expansion, we can simplify the equation to g(h) ≈ g_E (1 - (h / R_E)). This means that the gravitational acceleration decreases linearly with height as you go higher, enabling us to begin to predict how much lighter an object becomes as we ascend.

If you've ever climbed a hill, you may have noticed that it feels easier to lift a light backpack at the top than at the bottom. This feeling represents how gravity lessens with height. Here, the backpack becomes slightly lighter as you go higher because the force of gravity diminishes.

Now, consider a point mass m at a depth d below the surface of the earth. For the smaller sphere of radius (R_E - d), the point mass is outside it and the force on m is as if the mass of this smaller sphere is concentrated at the center:

$g(d) = \left(1 - \frac{d}{R_E}\right) g_E$.

When we talk about points located below the Earth's surface, the concept changes slightly. Although the mass outside the radius of the smaller sphere does not exert gravitational force on the point mass, the mass within the sphere does exert a force as if it was concentrated at the center. Hence, gravitational acceleration increases linearly as we go deeper into the Earth. The formula presented shows that g(d) falls to a lower value as d increases, illustrating that gravity is weakest at the core of the Earth.

Think of how a diver feels at different depths in water. Just like you can feel the pressure decrease as you swim to the surface, the force of gravity felt by firemen in a deep pit can similarly be thought of as diminishing—the deeper you go, the less gravitational pull you experience from the Earth above.

Thus, the acceleration due to Earth’s gravity is maximum at its surface, decreasing whether you go up or down.

This section summarizes the key takeaway: acceleration due to gravity is strongest at the Earth's surface, and it decreases both when ascending above the surface and when descending below the surface. The relationship described in this section introduces a unique characteristic of gravitational acceleration with elevation and depth.

Consider a trampoline: when you are at the lowest point, you feel a force pushing you upwards; as you rise, this force lessens. Gravity acts similarly. Think of climbing to the edge of the Earth, reaching lower depths, or even in a rocket—you will always feel gravity lessen as you travel away from or deeper into the Earth.

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