Question

How acceleration due to gravity varies with altitude? (increasing height from Earth's surfacw)

Answer 1

• Let the acceleration due to gravity ay Earth's surface be g

• Let height be h

• Let the acceleration due to gravity at a height h be gh.

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We know,

$g = \dfrac{GM}{ {R}^{2} }$

• G = Gravitational constant
• R= Radius of Earth
• M= Mass of Earth

Let's find the ratio of gh and g

$\dfrac{gh}{g} = \dfrac{ \dfrac{GM}{ {(R+ h)}^{2} } }{ \dfrac{GM}{ {R}^{2} } }$

→ $\dfrac{gh}{g} = \dfrac{ {R}^{2} }{ {(R + h)}^{2} }$

→$\dfrac{gh}{g} = {(1 + \dfrac{h}{R} )}^{ - 2}$

By Bi-nomial expansion we get,

$(1 + \dfrac{h}{R} ) \approx \: (1 - \dfrac{2h}{R} )$

→ Remember it's when R is less than or equal to 5% of Earth's radius

Thus we get,

$\red{\boxed{\bold{\large{gh = g(1 - \dfrac{2h}{r})}}}}$

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When h > 5% of radius of Earth:

$\red{\boxed{\bold{\large{ gh = \dfrac{GM}{{(R+ h)}^{2}}}}}}$

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• →Value of acceleration due to gravity decreases with increase in height.

Answer 2

Answer:

The acceleration of an object changes with altitude. The change in gravitational acceleration with distance from the centre of Earth follows an inverse-square law. This means that gravitational acceleration is inversely proportional to the square of the distance from the centre of Earth.