Variation of g — Core Principles

Core Principles

The acceleration due to gravity, 'g', is the acceleration experienced by objects due to Earth's gravitational pull, with an average value of $9.8,\text{m/s}^2$ at the surface. However, 'g' is not constant and varies significantly with location.

As we move to higher altitudes (height 'h' above the surface), 'g' decreases because the distance from the Earth's center increases. The formula is $g_h = g(1 + h/R)^{-2}$, which approximates to $g_h approx g(1 - 2h/R)$ for small 'h'.

Similarly, 'g' also decreases as we go deeper (depth 'd' below the surface) because only the mass of the Earth within the inner sphere contributes to the gravitational pull. The formula for depth is $g_d = g(1 - d/R)$, becoming zero at the Earth's center.

Finally, the Earth's rotation causes 'g' to vary with latitude ($lambda$). The centrifugal effect of rotation reduces the effective 'g' most at the equator ($lambda=0^circ$) and least (zero effect) at the poles ($lambda=90^circ$), given by $g' = g - Romega^2 cos^2lambda$.

Understanding these three primary variations is essential for NEET.

Important Differences

vs Variation of g with Altitude vs. Depth

Aspect	This Topic	Variation of g with Altitude vs. Depth
Formula (for small changes)	$g_h approx g(1 - 2h/R)$	$g_d = g(1 - d/R)$
Rate of decrease (for small changes)	Decreases approximately as $2gh/R$	Decreases approximately as $gd/R$
Maximum value	At Earth's surface ($h=0$)	At Earth's surface ($d=0$)
Minimum value	Approaches zero as $h o infty$	Zero at Earth's center ($d=R$)
Reason for decrease	Increased distance from Earth's center, weakening gravitational pull.	Decreased effective mass of Earth pulling the object (mass above cancels out).

The acceleration due to gravity 'g' decreases both with increasing altitude and increasing depth from the Earth's surface. However, the rate and underlying physical reasons differ. For small changes, 'g' decreases twice as fast with height as it does with depth. At infinite height, 'g' approaches zero, while at the Earth's center, 'g' becomes exactly zero. The altitude variation is due to increased distance from the entire Earth's mass, whereas depth variation is due to the reduction of the effective gravitating mass.