Variation of g with depth and height
Variation of g with Depth and Height
The acceleration due to gravity, denoted by ( g ), is not constant throughout the Earth's surface. It varies with altitude (height) above the surface as well as with depth below the surface. Understanding this variation is crucial for various scientific and engineering applications, such as geophysics, aerospace engineering, and GPS satellite operations.
Variation of g with Height
As we move away from the Earth's surface, the acceleration due to gravity decreases. The value of ( g ) at a height ( h ) above the Earth's surface can be calculated using the formula:
[ g_h = g_0 \left( \frac{R}{R + h} \right)^2 ]
where:
- ( g_h ) is the acceleration due to gravity at height ( h ),
- ( g_0 ) is the acceleration due to gravity at the Earth's surface (approximately ( 9.81 \, \text{m/s}^2 )),
- ( R ) is the radius of the Earth (approximately ( 6.371 \times 10^6 \, \text{m} )).
Example for Variation with Height
If we want to calculate the acceleration due to gravity at a height of 10,000 meters above the Earth's surface, we would use the formula:
[ g_{10000} = 9.81 \left( \frac{6.371 \times 10^6}{6.371 \times 10^6 + 10000} \right)^2 \approx 9.79 \, \text{m/s}^2 ]
Variation of g with Depth
The acceleration due to gravity also varies with depth below the Earth's surface. As we go deeper, the value of ( g ) decreases linearly assuming the Earth has a uniform density. The formula for ( g ) at a depth ( d ) is:
[ g_d = g_0 \left( 1 - \frac{d}{R} \right) ]
where:
- ( g_d ) is the acceleration due to gravity at depth ( d ),
- ( g_0 ) is the acceleration due to gravity at the Earth's surface,
- ( R ) is the radius of the Earth,
- ( d ) is the depth below the Earth's surface.
Example for Variation with Depth
To find the acceleration due to gravity 500 meters below the Earth's surface, we would calculate:
[ g_{500} = 9.81 \left( 1 - \frac{500}{6.371 \times 10^6} \right) \approx 9.81 \, \text{m/s}^2 ]
The change is very small because the depth is much smaller than the Earth's radius.
Table of Differences and Important Points
| Factor | Variation with Height | Variation with Depth |
|---|---|---|
| Formula | ( g_h = g_0 \left( \frac{R}{R + h} \right)^2 ) | ( g_d = g_0 \left( 1 - \frac{d}{R} \right) ) |
| Direction of Change | Decreases with increasing height | Decreases with increasing depth |
| Rate of Change | Changes with the square of the distance from the center of the Earth | Changes linearly with depth assuming uniform Earth density |
| Example Value | ( g_{10000} \approx 9.79 \, \text{m/s}^2 ) at 10 km above the Earth's surface | ( g_{500} \approx 9.81 \, \text{m/s}^2 ) at 500 m below the Earth's surface |
Conclusion
The acceleration due to gravity ( g ) varies with both height and depth. It decreases as one moves away from the Earth's surface, whether upwards or downwards. The rate of change with height is governed by an inverse square law, while the change with depth is linear (assuming uniform Earth density). Understanding these variations is essential for precise calculations in fields such as geology, engineering, and space exploration.