# Variation of g due to height and depth - comparative study with numerical (2023)

Last updated on April 23rd, 2022 at 02:38 pm

Variation of g with height and depth: Acceleration due to gravity or g varies as the height or depth varies with respect to the surface of the earth. This means the value of g on top of a mountain won’t be exactly the same as that on the earth’s surface. Similarly, g at a location considerably below the earth’s surface won’t be equal to the value of g on the earth’s surface. This is known as the variation of g with height and depth. Variation of g with height is expressed by the formula g1 = g (1 – 2h/R), where h<<R. And, the Variation of g with depth is expressed by the formula g2 = g (1 – d/R).

Here g1 is the acceleration due to gravity at a height of h with respect to the earth’s surface. And, g2 is the acceleration due to gravity at depth d with respect to the earth’s surface. R is the radius of the earth.

The extent of the variation of g with height differs from that of the variation of g with depth, but it’s to note that the value of g falls both with increasing height & with increasing depth, with respect to the earth’s surface. This also means the value of g is maximum on the surface of the earth itself.

Now, to discuss exactly how acceleration due to gravity changes with height and depth with respect to the surface of the earth, we will take the help of simple mathematics and analyze separately (1) the Variation of g with height and (2) the Variation of g with depth and derive the formulas describing this variation of g with altitude and depth.

[ Note: Here, the symbol g is used frequently and it means Acceleration due to gravity]

1. Variation of g with height | How does Acceleration due to gravity(g) change with height?
• Formula for g at height h
2. Variation of g with depth | How does Acceleration due to gravity(g) change with depth?
• Formula for g at depth d
3. Derive the Formula for acceleration due to gravity at height h | Variation of g with height derivation
4. Derive the Formula for acceleration due to gravity at depth d | Variation of g with depth derivation

## Variation of g with height | How does Acceleration due to gravity(g) change with height?

Variation of g 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), where h<<R. Here g1 is the acceleration due to gravity at a height of h with respect to the earth’s surface 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 meters from the surface of the earth becomes 9.7969 m/s^2. [ check with online calculator ]

### Formula for g at height h

The Variation of g with height is expressed by the formula g1 = g (1 – 2h/R), where h<<R. Here g1 is the acceleration due to gravity at a height of h with respect to the earth’s surface and R is the radius of the earth. This is the formula for g at height h.

## Variation of g with depth | How does Acceleration due to gravity(g) change with depth?

Variation of g with depth: As depth d increases below the earth’s surface the value of acceleration due to gravity falls. This is expressed by the formula g2 = g (1 – d/R). Here g2 is the acceleration due to gravity at depth d with respect to the earth’s surface and R is the radius of the earth.

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

(Video) Variation of g with height || Variation of acceleration due to gravity with height

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

### Formula for g at depth d

The Variation of g with depth is expressed by the formula g2= g (1 – d/R). Here g2 is the acceleration due to gravity at a depth of d with respect to the earth’s surface and R is the radius of the earth. This is the formula for g at depth d.

## Derive the Formula for acceleration due to gravity at height h|Variation of g with height derivation

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

We will derive the expression of g at a height h above the earth’s surface. This will show thevariation of acceleration due to gravity with height.

This section covers the variation of g with altitude. At a height of h from the surface of the earth, the gravitational force on an object of mass m is: F = GMm/(R+h)2

Here (R + h) is the distance between the object and the center of the earth.
Say at that height h, the gravitational acceleration is g1.
So we can write, mg1 = GMm / (R+h)2

=> g1 = GM/(R+h)2 _________________ (1)

Now we know on the surface of the earth, it is
g = GM / R2 [ see the proof: equation of g on earth’s surface ]

Taking the ratio of these 2,
g1/g = R2 /(R+h)2

= 1/(1 + h/R)2 = (1 + h/R)-2 = (1 – 2h/R) [ with the assumption that h<<R]

so, g1/g = (1 – 2h/R)

The Formula for the acceleration due to gravity at height h is represented with this equation:
=> g1 = g (1 – 2h/R) ______(2) [ with the assumption that h<<R]
g1 is the acceleration due to gravity at height h.

## Derive the Formula for acceleration due to gravity at depth d | Variation of g with depth derivation

Derivation of g2 = g (1 – d/R) – step by step proof

We will derive the expression of g at a depth d below the earth’s surface. This will show us the variation of acceleration due to gravity with depth.

Let’s say, a body of mass m is resting at point A, where A is at a depth of d from the earth’s surface.

The distance of point A from the center of the earth = R – d,
where R is the radius of the earth.

Mass of inner sphere = (4/3). Π. (R-d)3. ρ

Here ρ is the density. and Π is 22/7.

Now at point A, the gravitational force on the object of mass m is

F = G M m/ (R-d)2

= G. [(4/3). Π. (R-d)3. ρ] m/(R-d)2

= G. (4/3). Π. (R-d). ρ. m

Again at point A, the acceleration due to gravity (say g2) = F/m = G. (4/3). Π. (R-d). ρ _________________ (7)

Now we know at the earth’s surface, g = (4/3) Π R ρ G ( see the proof here: g formula on the surface of the earth using density)

Taking the ratio, again,

(Video) Numerical based on Variation of g with Altitude & depth || 11th CH08 ( Gravitation)-03

g2/g

= [G. (4/3). Π. (R-d). ρ ] / [(4/3) Π R ρ G]

= (R-d) / R = 1 – d/R.

=> g2 = g (1 – d/R)

The formula for the acceleration due to gravity at depth d is represented with this equation:
=> g2 = g (1 – d/R) ______ (8)
g1 is the acceleration due to gravity at a depth of d

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, [ with the assumption that h<<R]
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, with the assumption that h<<R.
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.

(Video) Variation of g With Height & Depth | Gravitation | Class 11

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

g2 = g (1 – d/R)

This gives the formula for g at depth d.
So as depth d increases below the earth’s surface, the value of g falls.

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

g2 = g (1 – d/R)
This gives the formula for g at depth d.
So at a depth d below the earth’s surface, the value of g falls by this amount: gd/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, with the assumption that h<<R

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

g2 = g (1 – d/R)
at a depth d below theearth’s surface

1) Now from the 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.

(Video) Variation of "g" due to Altitude ( Height )

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

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

Therefore the value of g at the center of the earth

is g (1- d/R) = g (1 – R/R)

=g (1-1) = 0

So we can see, that the value ofg 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 ofg will reduce by 1% with respect to that on the earth’s surface?

See Solution

Q3: If the value ofg at a small height h from the surface equals the value ofg 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

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## FAQs

### What is the variation of g with height and depth Numericals? ›

Variation of g with Height: The value of g is inverse to the height above the earth's surface; therefore, it decreases with increasing height. Variation of g with Depth: The value of g is inversely proportional to the depth below the earth's surface but directly proportional to the mass of the earth.

What is the formula for variation of g with depth? ›

The formula g2= g (1 – d/R) expresses the variation of g with depth. Here, g2 denotes the acceleration due to gravity at a depth of d from the earth's surface, and R denotes the earth's radius.

What is the formula for variation in g due to height? ›

Therefore gravity varies with height gh=g(1−2hR), and it decreases as we move above the surface of the earth.

What is the relation between height and depth for same g? ›

Updated On: 27-06-2022

Acceleration due to gravity at a depth d below the Earth's surface. Hence the acceleration due to gravity at a height h above the Earth's surface will be same as that of depth d = 2h, below the Earth's surface.

How do you calculate variation in depth? ›

We can calculate the variation of pressure with depth by considering a volume of fluid of height h and cross-sectional area A . Pa, which is atmospheric pressure. 2. PB = PT + gh.

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

Acceleration due to gravity decreases linearly with increase in depth.

What is variation the value of g? ›

This value is around 9.8 m/s2. But this value is not constant. This value varies with change in the latitude from the Earth's surface. The acceleration due to gravity (g) is large at the pole as compared to the equator.

How the value of g changes with the increase of depth? ›

As the depth increased the mass of the earth decreases. At the surface of earth this value will be maximum because radius will be maximum. When radius becomes less this value also decreases. Hence acceleration due to gravity decreases with increase in the depth.

Why does g value decrease with depth and height? ›

As the height increases from the surface of the earth , acceleration due to gravity decreases. Acceleration due to gravity depends also on the mass of the earth but it decreases with increase in height due to its inverse square relationship with distance.

How do you find g from height and time? ›

It only depends on height from the surface and the time period for which the body is flung.
...
1. h= \frac{1}{2}gt^2.
2. v²= 2gh.
3. v=gt.

### What is the ratio of value of g at height? ›

The value of the acceleration due to gravity is at a height h = R 2 ( = radius of the earth) from the surface of the earth. It is again equal to at a depth below the surface of the earth. The ratio equals : 4 9.

How do you show the value of g decreases with height? ›

The value of 'g' decreases with altitude because gh = g (1 + h/R)-2 [where gh is the acceleration due to gravity at height h]. This formula suggests that 'g' decreases with altitude.

Does the value of g change with depth and height? ›

At the surface of the earth this value will be maximum because R will be max. When R becomes less ( i.e when depth increases) this value also decreases. Hence, acceleration due to gravity decreases with increase in depth.

What happens to the value of g if we go up to the height of two earth radiuses? ›

For instance, if an object were moved to a location that is two earth-radii from the center of the earth - that is, two times 6.38x106 m - then a significantly different value of g will be found. As shown below, at twice the distance from the center of the earth, the value of g becomes 2.45 m/s2.

What is the variation of height and depth? ›

This is referred to as g variation with height and depth. The formula g1 = g (1 – 2h/R) expresses the variation of g with height. The formula g2 = g (1 – d/R) expresses the variation of g with depth.

What are the 3 ways to measure variation? ›

Above we considered three measures of variation: Range, IQR, and Variance (and its square root counterpart - Standard Deviation). These are all measures we can calculate from one quantitative variable e.g. height, weight.

What is one numerical on Newton's universal law of gravitation? ›

It's a general Physics law derived from the observations by Isaac Newton. In modern language, the law states: every point mass attracts another single point mass by a force pointing along the line intersecting both the ends. G = universal gravitational constant = 6.67259 x 1011 N m2/kg2.

What is numerical on variation of g? ›

G = universal gravitational constant (6.67×10-11 Nm2/kg2) m = mass of the object, M = mass of the earth, r = radius of the earth.

What are the factors affecting the variation in value of g? ›

The factors influencing the value of g are the earth's shape, altitude, and depth under the earth's surface. The earth is not spherical. It is slightly straight or flat at the poles and swelling in the tropics.

What are the 3 factors on which variation in value of g depends in short? ›

Therefore, acceleration due to gravity (ggg) depends on the mass of the planet, distance between centre of planet and object, shape of the planet, rotational parameters of the planet.

### Does g increase or decrease with height? ›

Earth's gravity depends on the distance between the centre of the earth and the object on the surface of the earth. As the altitude increases, the distance from the centre of the earth increases. Therefore, the gravity decreases as it the inversely proportional to the distance.

What is the relation between height and gravity? ›

gravity increases with height. gravity is significantly less on high mountains or tall buildings and increases as we lose height (which is why falling objects speed up) gravity is caused by the Earth spinning. gravity affects things while they are falling but stops when they reach the ground.

What are two reasons why g varies on the surface of the earth? ›

Changes with time. The gravitational potential at the surface of Earth is due mainly to the mass and rotation of Earth, but there are also small contributions from the distant Sun and Moon. As Earth rotates, those small contributions at any one place vary with time, and so the local value of g varies slightly.

How do you calculate g value? ›

G is the universal gravitational constant, G = 6.674 x 10-11 m3 kg-1 s-2. M is the mass of the body measured using kg. R is the mass body radius measured by m. g is the acceleration due to the gravity determined by m/s2.

How do you calculate your g? ›

Subtract initial velocity from final velocity. Divide the difference by time. Divide the resultant by the acceleration due to gravity, 9.81 m/s², to obtain the g force value.

How do you find the value of G at a height from earth? ›

The value of g at a certain height h above the free surface of the earth is x/4 where x is the value of g at the surface of the earth.

At what height g decreases 36% of its value? ›

Hence, height is one fourth of earth radius, 1600km.

What is the calculation of g? ›

G is the universal gravitational constant, G = 6.674 x 10-11 m3 kg-1 s-2. M is the mass of the body measured using kg.

What is the variation of g with latitude notes? ›

Variation of g with latitude

From the above expression, we can infer that at the equator, λ = 0, g' = g – ω2R. The acceleration due to gravity is minimum. At poles λ = 90; g' = g, it is maximum. At the equator, g' is minimum.

How do you find the experimental value of g? ›

y = v0t − gt2 2 . So if the initial velocity is known, the time a projectile takes to travel through the air will provide infor- mation about the value of g.

### What is the variation value of g? ›

Variation of the value of gravitational acceleration due to rotation of Earth. Equation (3) gives us the value of acceleration caused by gravity on the surface of Earth. This value is around 9.8 m/s2.

How does g vary with latitude and elevation? ›

This means that acceleration due to gravity varies with height and decreases as we go to higher altitudes above the surface of earth. Latitudes represent points on the same altitude and therefore, there is no change in acceleration due to gravity with change in latitude.

What is variation of g? ›

g' = GM R e h. Mass at a height h from the center of the Earth. g' = GM R e h R e.

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