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Understanding Gravity Reduction Above Earth's Surface

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The force exerted by gravity is inversely proportional to the square of the distance from the Earth's center of gravity, following the inverse square law. This relationship is crucial for understanding how the acceleration due to gravity changes at different altitudes above the surface of the Earth. This article explores this phenomenon in detail, with particular attention to the points where gravity is reduced to 19% of its surface value, or half of its surface value, providing insights through mathematical derivations and examples.

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General Formula for Gravitational Acceleration

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The acceleration due to gravity (a) at a height (h) above the Earth's surface can be described by the equation:

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a GM/r^2 where a is the acceleration due to gravity at height h, G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth.

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Calculating Gravity Reduction at Specific Altitudes

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Given that the gravitational acceleration at the Earth's surface (gsurface) is given by:

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gsurface GM/R^2

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Where R is the mean radius of the Earth. We can express the gravitational acceleration at a height h above the Earth's surface as:

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a gsurface * (R / (R h))^2

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At a height of 19% reduction (i.e., 0.19 * gsurface):

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0.19 * gsurface gsurface * (R / (R h))^2

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Solving for h:

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0.19 (R / (R h))^2

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1.4142 ≈ R / (R h)

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(R h) ≈ R / 1.4142

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h ≈ R - (R / 1.4142) ≈ 0.3858R ≈ 0.3858 * 6371 km ≈ 2450 km

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Thus, to experience 19% reduction in gravitational pull, one must be approximately 2450 km above the Earth's surface.

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Half of Surface Gravity

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Calculating the altitude where the gravitational acceleration is half of the surface value (0.5 * gsurface):

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0.5 * gsurface gsurface * (R / (R h))^2

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Solving for h:

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0.5 (R / (R h))^2

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1.4142 ≈ R / (R h)

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(R h) ≈ R * 1.4142

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h ≈ R * 1.4142 - R ≈ R * 0.4142 ≈ 6371 km * 0.4142 ≈ 2642 km

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Therefore, to experience half of the surface gravitational pull, one must be approximately 2642 km above the Earth's surface.

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Graphical Representation

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A graphical representation of the gravity curve shows that at Earth's radius (R), the gravity is approximately 9.8 m/s2. At an altitude of 2R, the gravity is approximately a fourth of its value at the surface (2.45 m/s2). At an altitude of 8919.4 km (approximately 1.414 * R), the gravity is approximately half of the surface value (4.9 m/s2).

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Conclusion

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This article has provided the mathematical derivations and practical insights into the reduction of gravitational acceleration at different altitudes above the Earth's surface. Whether analyzing for theoretical purposes or practical applications like space travel, understanding these principles is crucial.