Gravitation-Variations

In the 16th century, Johannes Kepler discovered three laws that define the movement of planets in the solar system. The same develops this idea: the motion of satellites such as the moon around the earth.

Kepler’s law of planetary motion

The Law of Orbits, or Kepler’s First Law of Planetary Motion

The orbit of every planet around the sun is an ellipse. The centre of the sun is found at one of the foci in the planet’s elliptical orbit. The sun is in a single point of focus, and all the planets travel in an elliptical orbit.

This means that the distance between the sun and the planet changes as the planets move.

The point at which a planet is closest to the sun is called the perihelion. The point at which the planet is found at its furthest distance from the sun is called the aphelion.

The elliptical shape of orbit allows us to find the distance between two focal points in which the planet travels. This is also why planets have different seasons.

The Law of Equal Areas, or Kepler’s Second Law of Planetary Motion

The imaginary line that joins both the planet and sun sweeps an equal area of space in equal intervals of time. Eventually, the planets do not move at a constant speed along the orbits as believed to be. Instead, their speed varies with the equal area of space swept by the planet in a particular time limit.

The highest speed is observed at the perihelion, while the planet moves slower at the aphelion.

If r = the distance from the sun, then perihelion is rmax, and aphelion is rmin.

rmin + rmax =  2 a  (length of major axis of ellipse)

The confirmation of this law can be obtained by using the formula of angular momentum, 

L =  mr2ω.

We will arrive at: dA / dt = constant.

From the above equation, it can be deduced that the area swept in an equal interval of time is a constant value.

The Law of Periods, or Kepler’s Third Law of Planetary Motion

The cube of a planet’s semi-major axis is directly proportional to the square of its time period of revolution around the sun in an elliptical orbit 

This may be represented by the formula:

T2 ∝ a3

According to Kepler’s third law, the period of a planet’s orbit around the sun increases when the radius of the orbit is expanded. Thus, the shorter the planet’s orbit around the sun, the faster it takes to complete one revolution. 

Kepler’s third law is more generalised using the equations of Newton’s law of gravitation and laws of motion:

T2 = 42G(M1+M2)a3

As a result, it can be deduced that Mercury, the planet closest to the sun, orbits it in only 88 earth days. We all know that the earth takes 365 days to complete one revolution around the sun, while Saturn takes 10,759 days.

Free fall:

Free fall is the descent of a body or any specific item from such a height and toward the earth under the effect of gravity where no external or internal force is exerted on it.

So take into account that if you release two items of different weights from the very same height, they both will tend to fall at the very same time. As a result, the acceleration of a particular object descending freely to the earth is irrespective of the object’s mass and not dependent on the mass of the item.

When a feather and a coin are both dropped from the same height, which will come down faster?

According to the universal law of gravitation, the coin and the feather will tend to fall at the very same time, but the feather will take a lot of time to arrive due to more air resistance acting on it. This indicates that if you perform this experiment in a mere vacuum, both will tend to fall at the same moment.

Acceleration owing to gravity is the homogeneous acceleration developed from a freely falling body caused by the gravitational force available on the earth.

Gravity-induced acceleration variation_ariation of ‘g’:

Gravity’s acceleration changes depending on where you are on the planet. The radius fluctuates because the earth is perfectly not spherical. It is less at the poles and higher near the equator. And g increases as the corresponding radius decreases near the equator. Inside the earth, the equation does not support. As we go up or inside the earth’s surface, the value of g drops.

Here is the table showing the gravitational constant & Altitude represented in Metre.

Conclusion

Kepler’s laws of planetary motion are laws that explain the motions of the bodies in the solar system in astronomical and mediaeval physics. Johannes Kepler, a German astronomer, developed them. The following are Kepler’s three laws of planetary motion: 

(1) All planets move in elliptical circular orbits, with the Sun as being one of the foci. 

(2) A radius vector connecting any planetary to the Sun sweeps away equivalent regions over similar time periods.