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Basic Mechanics: Gravity and Newton's Law of Gravitation

Mohammad Yasir is a physics graduate from the University of Delhi and currently enrolled in the master's programme at IIT.

Demonstration of the gravitational force between two objects.

Demonstration of the gravitational force between two objects.

Introduction

Gravity is one of the most interesting and widely tackled problems in mechanics that has applications in every single problem you can imagine. It is a force exerted by all objects in the universe on all other objects. The reason it is so ubiquitous in physics is simple: it is ubiquitous in the universe as well. Even in space, there are billions of celestial objects scattered helter-skelter and thus, all calculations related to worlds beyond this world require gravity to be considered.

If you pursue advanced studies in fields that involve gravitation, you will find yourself going down a rabbit hole of field equations, general relativity, space-bending, and whatnot. However, this rabbit hole starts with one very simple equation: that of Newton's Law of Gravitation.

Portrait of Sir Isaac Newton at 46 by Godfrey Kneller, circa 1689.

Portrait of Sir Isaac Newton at 46 by Godfrey Kneller, circa 1689.

Quick History Lesson

Before we dive in, here's a small paragraph about the history of how gravity has been discussed in scholarly texts since the time of the ancient Greeks.

During the 16th century, while there was no expression to dictate how gravitational force interacted with matter, a large number of theories still came into being, most of which were part of the Scientific Revolution and laid the foundation for modern-day science. In particular, Galileo proposed that the speed of a falling object increased as the square of time elapsed (correctly so), and later, he performed the famous Leaning Tower of Pisa experiment, which demonstrated how the rate of falling was the same for objects of differing masses.

Finally, it was around 1684 that Newton published his book titled the Mathematical Principles of Natural Philosophy, which explained how Kepler's laws could be physically explained via the concept of gravitational force. It is this law that we are going to discuss in this article, and we will also derive a few commonly used values.

Newton's Law of Gravitation

First proposed in Newton's Principia, this famous law was a ground-breaking result that laid the foundation of a large number of modern theories. It states that given two objects of masses m₁ and m₂, separated by a distance of r units, the force of gravity exerted by these objects on each other is:

  • directly proportional to the product of the masses (m₁m₂)
  • inversely proportional to the square of the distance between them.

Combining the above two factors and multiplying a constant of proportionality yields us the result attached here.

Mathematical expression of Newton's law of gravitation.

Mathematical expression of Newton's law of gravitation.

The direction of gravitational force is directed across the line joining the centers of mass of the two objects in question and, pursuant to Newton's third law of motion, the force exerted by m₁ on m₂ is equal and opposite to the one exerted by m₂ on m₁.

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Some Common Expressions

As far as basic mechanics is concerned, the above information about Newton's law of gravitation will suffice for a basic understanding. However, the formulae and concepts that emanate out of the above expression, especially when the source of the gravitational force in question is Earth merit separate mention even in this short text. I will try to derive a few common quantities and formulae for Earth using the above equation. We will start with the simplest one: the acceleration due to gravity on Earth.

I have mentioned the centrifugal term in the definition above. However, in most scenarios, this effect is so small that it is of no consequence for our discussion. Regardless, a little extra knowledge is never harmful.

The easiest way to calculate the acceleration due to gravity is via Newton's second law. We can equate the general formula for force F = ma with the gravitational force to find the value of a. Take a look

newton-law-gravitation-gravity-gravitational-force

The above value is an excellent estimate of the acceleration due to gravity on Earth. The standard value accepted internationally is 9.80665 m/s². In most numerical calculations where extreme accuracy isn't desired, the value g = 9.8 m/s² is commonly used. The symbol g is a common way of representing the acceleration due to gravity of the Earth. Note that due to the change in gravitational force with distance, this value changes as we change our vertical position from the Earth's center. I will not go into much detail about that here since these cases are rarely used at the high school level.

Escape velocity is another common term used in the study of gravitation of a planet. Interestingly enough, the expression for escape velocity does not include the mass of the object in question; rather, it depends solely on the mass of the planet one is escaping from.

To derive an expression for this often-used quantity, all we need to do is equate the kinetic energy of an object with its potential energy at a distance equal to the radius of the earth. This is a natural enough step to take since it allows us to calculate the kinetic energy (or equivalently, the velocity), which would allow an object to just escape the Earth's gravitational pull. Calculations yield

newton-law-gravitation-gravity-gravitational-force

Notice how in the definition I used the term unpropelled object. This simply means that the object in question has no way of giving it any additional push upwards. Earth's escape velocity is 11.1 km/s, which means that if an object were to start moving vertically upwards with a velocity of 11.1 km/s, it would be able to just escape Earth's gravitational pull before coming to a halt due to the retardation it experience on account of Earth's gravity.

Practically, on account of air resistance and other factors, there would be infinitesimal, almost negligible changes in the value of the escape velocity we just obtained. But for most scenarios, the above calculations suffice.

Conclusion

Newton's law of graviation is of immense importance right off the bat when it comes the realm of physics. While it appears quite simple, its applications and implications are almost impossible to quantify. Even though Albert Einstein's general theory of relativity supersedes this law, Newton's contributions are still valid in almost all practical scenarios save for where there are enormously dense objects like blackholes.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Mohammad Yasir

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