In this post, we will discuss in detail the following: (1) Variation of g with height and (2) Variation of g with depth. (g here stands for the acceleration due to gravity)



So basically we will see how acceleration due to gravity changes with height and depth with respect to the surface of the earth.



To add here that as we gain altitude or height with respect to earth’s surface we find a variation of g with height (g value falls with increasing height). Similarly deeper below the earth’s surface, we notice a variation of g with depth (g value falls with increasing depth).

Let’s find out the formula for g at height h and the formula of g at depth h in the following sections.

Variation of g with height

Variation of g with height or How Acceleration due to gravity changes with height?



As altitude or height h increases above the earth’s surface the value of acceleration due to gravity falls. This is expressed by the formula g1 = g (1 – 2h/R). Here g1 is the acceleration due to gravity at height h and R is the radius of the earth.



So at a height h above the earth’s surface, the value of g falls by this amount: 2gh/R.

For example, considering g = 9.8 m/s^2 on the earth’s surface, g1 at a height of 1000 meter from the surface of the earth becomes 9.7969 m/s^2. [ check with online calculator ]

Variation of g with depth

Variation of g with depth or How Acceleration due to gravity changes with depth from the earth’s surface?



As depth h increases below the earth’s surface the value of acceleration due to gravity falls. This is expressed by the formula g2 = g (1 – h/R).

Here g2 is the acceleration due to gravity at depth h and R is the radius of the earth.



So at a depth h below the earth’s surface, the value of g falls by this amount: gh/R

For example, considering g = 9.8 m/s^2 on the earth’s surface, g2 at a depth of 1000 meter from the surface of the earth becomes 9.7984 m/s^2. [ check with online calculator ]

Formula Flashcard– A quick reckoner for readers in a hurry.

Derivation of all related formulas



1> First we will derive the expression of Acceleration due to gravity (g) on the earth’s surface. We need these expressions to carry on the next step given below.



[ Derivation of g formulas on the earth’s surface]



2> Then we will derive the expression of g at a height h above the earth’s surface. This will show the variation of acceleration due to gravity with height.



[ Derivation of g1 = g (1 – 2h/R) – step by step proof]



3> After this, we will derive the expression of g at a depth h below the earth’s surface. This will show us the variation of acceleration due to gravity with depth.



[ Derivation of g2 = g (1 – h/R) – step by step proof]



What is the variation in acceleration due to gravity with altitude?

g1 = g (1 – 2h/R)

This gives the formula for g at height h.

So as altitude h increases, the value of acceleration due to gravity falls. This describes the variation of g with height or altitude.

What is the change in the value of g at a height h?

g1 = g (1 – 2h/R)

This g1 gives the formula for g at height h.

So the value of g will decrease by this amount: 2hg/R, at a height h above the earth’s surface. R is the radius of the earth.

What is the variation in acceleration due to gravity with depth?

g2 = g (1 – h/R)



This gives the formula for g at depth h.

So as depth h increases below the earth’s surface, the value of g falls.

What is the change in the value of g at a depth h?

g2 = g (1 – h/R)

This gives the formula for g at depth h.

So at a depth h below the earth’s surface, the value of g falls by this amount: gh/R. R is the radius of the earth.

In the next paragraph, we will compare these 2 equations to get a clearer picture.

How Acceleration due to gravity changes with height and depth? Equations – formula- Comparison

Let’s summarize how acceleration due to gravity changes with height and depth.

The formula for the acceleration due to gravity at height h (showing Variation of g with altitude)

g1 = g (1 – 2h/R)

at a height h from the earth’s surface

The formula for g at depth h (showing Variation of g with depth)

g2 = g (1 – h/R)

at a depth h below the earth’s surface

1) Now from equations, we see that both g1 and g2 are less than g on the earth’s surface.



That means acceleration due to gravity is maximum at the surface of the earth.



2) We also noticed that, g1 < g2

And that means:

1) the value of g falls as we go higher or go deeper.



2) But it falls more when we go higher.



3) It is also clear that acceleration due to gravity is maximum at the earth’s surface.

What is the value of acceleration due to gravity at the center of the earth?

At the center of the earth, depth from the earth’s surface is equal to the radius of the earth.

So we have to put h = R in the equation g2 = g (1 – h/R)



Therefore the value of g at the center of the earth



is g (1- h/R) = g (1 – R/R)



=g (1-1) = 0

So we can see, the value of g at the center of the earth equals zero.

Numerical Problems (based on the variation of g)

Q1: What is the value of g at a height 4 miles above the earth’s surface? The diameter of the earth is 8000 miles. g at the surface of the earth = 9.8 m/s2

See Solution

Q2: At what depth under the earth’s surface, the value of g will reduce by 1% with respect to that on the earth’s surface?



See Solution

Q3: If the value of g at a small height h from the surface equals the value of g at a depth d, then find out the relationship between h and d.

See Solution

Variation of g with height and depth – how g changes with height and depth