The Mathematics of Gravity: Everything We Do and Don't Know

What is gravity?   

Gravity is a natural force that that attracts a body towards the center of the earth, or towards another physical body which has mass. If an object has mass it also has gravity, and the extent to which an object exerts gravity on other objects depends directly on its own mass. However, gravity also has an inverse relationship with distance, so that the force weakens with the increase in distance between objects.   

There are four fundamental forces found in nature that control all natural interactions: the strong nuclear force, electromagnetic force, weak nuclear force, and gravitational force. Gravity, which is the weakest force, can not influence interactions at the subatomic level but it is a dominant force in the cosmic realm. It plays a key role in the formation, path, and behavior of planets, asteroids, stars, solar systems, etc.  

Difference between mass and weight

If you weigh 150 lb (68 kg) on earth then your weight would be 25.5 lb (11.5 kg) and 379.5 lb (172 kg) on Moon and Jupiter respectively. How is this possible? This is because weight is the product of your mass and the force of gravity. 

W = mg

here, 

W = weight

m = mass, g = gravity

(m = volume x density)

Mass is a measurement of matter, it can never be zero for any given object. The SI unit of mass is the kilogram (kg), but weight is measured in Newton (N) and it can be zero for an object, for example, if the object is in a zero-gravity environment.

For example, suppose that you have a mass of 100 lb (on earth, your weight will also be 100 lb, or 980 N) on earth, when you go to the moon, your weight will be only around 17 lb. In this case, it is not that your weight has decreased between the Earth and the Moon. On Earth your mass and weight are effectively the same, but on the Moon your weight (which takes into account the effect of gravity) decreases because there is only 1/6th the gravity on the Moon compared to Earth. 

Mass (m) = 100 kg
Weight_earth = 100 x g (which is equal to 9.8 m/s²)

Weight_earth  = 980 N 

Weight_moon = m x g_moon 

since gravity on the moon is ⅙ of earth’s gravity 

Weight_moon = 100 x 9.8 x1/6 (mass is still 100 kg) 

Weight_moon = 163.3 N (which is equal to 16.65 lb)

The weight of a body varies depending on the force of gravity. It is a vector quantity, having both magnitude and direction. In contrast, mass is a scalar quantity, as it has only magnitude.

Newton’s law of gravitation

In was Sir Isaac Newton who put forward the concept of gravitational force in his 1687 treatise Philosophiæ Naturalis Principia Mathematica. In it, he calculated relative gravitational force. According to Newton’s universal law of gravitation, the attractive force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

F ∝ m_{1 ,} F ∝ m₂ 

F ∝ 1/r²

F = Gm₁m₂/r²

where,
F = gravitational force exerted by the earth on an object of mass m₁, 
m₁ = mass of an object
m₂ = mass of earth (5.98 × 10²⁴ kg)
G = gravitational constant (6.67408 × 10^-11 m³ kg^-1-s^-2)
r = radius of earth (6.38 × 10⁶ m)

We know that F = m₁g 

 m₁g  =  Gm₁m₂/r² 

g = Gm₂/r²

 g = (6.67408 × 10^-11 m³ kg^-1-s^-2).(5.98 × 10²⁴ kg) / (6.38 × 10⁶ m)²

g = 9.8 m/s²

According to Newton, g is defined as the acceleration due to gravity near the Earth’s surface.

Variation in Earth’s gravity

Earth is a solid sphere that is always rotating around the sun, and the movement and motion of earth also affects the value of gravitational acceleration on an object in different situations.

Change in value of g with height

Since gravity shares an inverse relation with distance, an increase in the height of an object from the surface of the earth causes a decrease in the value of g, and vice versa. 

The following formula depicts the variation in the acceleration due to Earth's gravity with height:

g_h = g[R/(R+h)]²

here,
g_h = acceleration due to gravity at height h above the sea level.
R = radius of the earth 
g = standard gravitational acceleration

When an object is present at an infinite distance from earth, g comes out to be zero.

Change in value of g with latitude 

Earth is around 43 km wider at the equator than pole-to-pole, and this difference between the earth’s circumference and diameter is called the equatorial bulge. This is caused by the centrifugal force produced due to the rotation of the earth on its axis. 

Due to this naturally occurring phenomenon, the earth has the shape of an oblate spheroid rather than a perfect sphere. This oblateness of our planet means that its center of gravity is a bit closer to the poles and farther from the equator.In fact, If you stand at sea level on the equator, you are 6378 km from the center of the earth, while at each pole, you are only 6357 km from the center of the earth. 

Because gravity weakens the farther away you get from a gravitational body, the points on the equator have weaker gravity than the poles. In other words, acceleration due to gravity is at a maximum near the poles and at a minimum near the equator.

The combination of the equatorial bulge and the centrifugal force due to rotation mean that sea-level gravity increases from about 9.78 m/s² at the Equator to about 9.83 m/s² at the poles. This means that an object will weigh a small amount more at the poles than at the Equator.

The latitude of a point is the angle (θ) between the equatorial plane and the line joining that point to the center of the earth. The latitude of the equator is 0° and that of the poles is 90°. Then, if we consider a body, m, with a mass P and latitude θ on Earth's surface. Then, let gθ be acceleration due to gravity at point P.

Due to the rotation motion of the Earth around its axis, the body at P experiences a centrifugal force of mrω²cosθ. The body is acted on by two forces - it's weight, mg, drawing it toward the center of the Earth, and mrω²cosθ, acting outward. The difference between the two forces gives the weight of the body at that point.

                                       mgθ = mg - mrω²cosθ

       cosθ = distance of point P from Earth's axis / radius of the Earth

                 = r / R

                                    therefore, r = R cosθ

       substituting this in to the original equation gives us:

                                 mgθ = mg - m(R cosθ)ω²cosθ

     and,

                                    gθ = g - Rω²cos²θ

here,
gθ = gravity at a given latitude
ω = earth’s angular velocity
mrω = centrifugal force
R = radius of the Earth
r = distance of point P from Earth's axis
g = standard gravitational acceleration 

For the poles, θ = 90°, so

 gθ = g

At the equator, θ = 0°, so

 gθ= g – Rω²

The centrifugal force is proportional to the tangential speed of the rotating reference frame. Since centrifugal force points outwards from the center of rotation, it tends to cancel out a little bit of earth's gravity. Because the equator is moving quickly as the earth's spins, it has a lot of centrifugal force. In contrast, the poles are not spinning at all, so they have zero centrifugal force.

Change in value of g with depth

The value of g decreases when an object travels deep inside the earth, at the center of the earth, the acceleration due to gravity becomes zero, but on the earth’s surface gravity is found to be maximum.

If an object of mass (m) travels to distance (d) below the earth’s surface, then the acceleration due to gravity at d depth (gd) can be obtained by taking the value of g in terms of density (ρ).

g = Gm/R²

Now, let ρ be the density of the material of the Earth, and

mass = volume x density

M = 4/3 πR³ x ρ

Now at depth ‘d’, the acceleration due to gravity is given by;

g_d = 4/3 × πG (R – d)ρ

Solving the equation further, we get

g_d = g(1-d/R)

If an object reaches the center of the earth, then d = R, and there will be no acceleration due to gravity at the center of the Earth.

Interesting facts about gravity

 There are various shocking and hidden aspects in which the gravitational force affects our life.

• Bones are not unchanging — they constantly reshape themselves in relation to the stresses that are put on them. As with muscles, if you don't use your bones by moving around under pressure, they will weaken. Bone loss occurs in the weightless environment of space because bones no longer have to support the body against gravity. A study from NASA indicates that astronauts can lose up to 1% of their bone mass for each month they spend in space. Once astronauts return to earth, the bones take some time to regain their strength. Blood pressure, which has equalized throughout the body while in space, also needs some time to return to a normal, Earth-bound, pattern where the heart must work harder to circulate the blood.
• Plants also grow differently in a low-gravity environment. On Earth, the starch grains present in plant roots sink towards the ground, due to the effects of the gravitational pull, and this helps guide the downward movement of plant roots. Research done on the ISS has shown that, while the roots still grow away from a light source (as they do in soil on the Earth), the roots responded to the lack of gravity by taking a straighter path through the growth medium, and by curving less. 
   
• NASA uses its GRACE satellites to measure changes in the rate at which plants and the land surface release moisture into the air. These processes are collectively known as evapotranspiration, and NASA has calculated its increase by using observations from gravity satellites. By gauging the mass change of water between the oceans and the continents, NASA researchers determined that the rate of increase in evapotranspiration has increased by about 10% since 2003, due to global warming. This increase is important, as evapotranspiration is critical to the global water cycle, which ultimately creates the conditions for life on land.
  

  

• A graviton is a hypothetical quantum of gravitational energy and is regarded as a particle. Although gravitons have never been observed, we know that if they exist they would have zero mass and travel at the speed of light. Gravitons have not been found yet because gravity is so incredibly weak that a single graviton would be too weak to detect. However, scientists also theorize that other types of 'exotic' gravitons might exist which could be detectable in other dimensions. Using particle accelerators like the Large Hadron Collider, physicists are searching for these extra dimensions, in part by looking for the expected decay products of massive gravitons. They haven’t found anything yet, but they are continuing to look.
  

Gravity is an essential phenomenon that affects both the cosmos and the human world, from the way plants grow to the actions of bodies in space, gravity affects almost everything that happens around us and beyond.