Gravitation Definition
In 1665, the concept of gravitation was put forth by Sir Isaac Newton when he was sitting under the tree, an apple fell down from that tree on the earth.
This sparked an idea in his mind that all bodies are attracted towards the center of the earth where he said that gravitation is the force of attraction between any two bodies separated from each other by a distance.
This concept played a major role in the initiating birth of stars, controlling the entire structure of the universe.
At present, this concept has significant applications in the advancement of physics.
Gravitational force is the weakest force among all the basic forces of nature.
Universal Law of Gravitation
In 1687, an English mathematician and physicist, Isaac Newton put forward this law to explain the observed motions of planets and their moons.
Newton’s law of universal gravitation states that any particle of matter in the universe attracts another one with a force varying directly as the product of the masses and inversely as the square of the distance between them.
Consider two bodies A and B of mass m1 and m2 separated by a distance r such that the force of attraction acting on them are represented as shown in the figure below:
In figure.1, the two bodies having forces of attraction F1 and F2 have a tendency to move towards the center of gravity..
Such that F1 = F2.
Therefore, the universal law of gravitation formula is given by,
F ∝ m1 . m2 / r2
Or
F = \[\frac {G(m_1.m_2)} {r^2}\].... (1)
Here, G is called the Universal gravitational constant (a scalar quantity).
The value of G remains constant throughout the universe and is independent of the nature and size of the bodies.
Definition of G
When m1 = m2 = 1 and r =1
Then from eq (1)
F = G
It says that the magnitude of the attractive force F is equal to G, multiplied by the product of the masses and divided by the square of the distance between them.
State Two Applications of Universal Law of Gravitation
Newton’s law of gravitation holds good for objects lying at very large distances and at short distances as well.
It fails when the distance between the two bodies is less than 10−9 m.
There are various applications where Newton’s law , two of them are discussed below:
The predictions about the orbits and time period of the modern artificial satellites made on the basis of this law proved to be very accurate.
The prediction about solar and lunar eclipses, made on the basis of this law, came out to be very true.
Importance of Universal Law of Gravitation
The gravitational force of earth ties the terrestrial objects to the earth.
This law explains the attractive force between any two objects having a mass.
The formation of tides in the ocean is due to the force of attraction between the moon and ocean water.
All planets make an elliptical revolution with the sun.
The rotation of the earth around the sun.
The rotation of the moon around the earth.
Derivation of Universal Law of Gravitation
This law states that any two objects pull on each other with force gravity.
Newton’s law brought up the new concept where he said:
Total force acting on an object = object’s mass x object’s acceleration
Total force is the force of gravity or Fg.
So,Fg (gravity force pulling on object) ∝ object’s mass (m)
The earth pulls the object towards itself.
The mass of earth = M and gravity force = Fg
So, Fg (gravity force) ∝ Earth’s mass (M)
Planets move around the sun in an elliptical orbit because gravity force provides the net centripetal force pulling the planet towards the center of its circle given by
Fg = \[\frac {m} {r v_2}\] …(2)
Since moon orbits the circumference of the circle in one period given by
velocity, v = \[\frac {2\Pi r} {T}\]
putting in eq(2)
Fg = \[\frac {m} {r}\].
\[\frac {2\Pi r} {T}^2\]
On solving,
Fg = 4
\[\frac {\Pi 2 mr} {T_2}\]
Multiplying both the sides by
\[\frac {T_2} {r}\]
we get
Fg.
\[\frac {T_2} {r}\] = \[\frac {4\Pi 2 mr} {T_2}\] . \[\frac {T_2} {r}\]
\[\frac {T_2} {r}\] Fg = 4 π2 m …(3)
Since Fg ∝ r2
we get F = k r2
putting it in eq(3)
We get that
\[\frac {T_2} {r(k/(r^2))}\] = 4 π2 m π2m
equivalent to the equation of kepler’s third law
i.e., \[\frac {T_2} {r^3}\] = constant (Newton considered Fg ∝ 1/ r2
Therefore, we inferred that
Fg ∝m \[\frac {M_1} {r^2}\]
Combining these three terms we get,
Fg ∝ \[\frac {mM} {r^2}\]
Removing this proportionality constant we get
F = \[\frac {G(MM)} {r^2}\] and G = \[\frac {F.r^2} {(m_1.m_2)}\].
Story Behind Law of Gravitation :
Once upon a time, Isaac Newton saw an apple fall from the tree. Looking at the motion of the apple while it was falling, he wondered if the same force would work on the moon as well. Drowning deep in this theory, he decided about 'why did this apple fall on earth but not on the moon'? Then, newton realized the forces acting on the falling objects otherwise they would not move from the position of rest.
FAQs on Universal Law of Gravitation
1. What is Newton's Universal Law of Gravitation in simple terms?
Newton's Universal Law of Gravitation states that every object in the universe attracts every other object with a force. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Simply put, heavier objects pull harder, and the pull gets weaker as objects get farther apart.
2. What is the formula for the Universal Law of Gravitation and what do its variables mean?
The formula for the Universal Law of Gravitation is: F = G(m₁m₂)/r². Here is what each variable represents:
- F is the gravitational force between the two objects.
- G is the universal gravitational constant.
- m₁ and m₂ are the masses of the two objects.
- r is the distance between the centers of the two objects.
3. Why is the law of gravitation called 'universal'?
The law is called 'universal' because it applies to all objects with mass, anywhere in the universe. It doesn't just apply to objects on Earth. The same law governs the force that causes an apple to fall from a tree, keeps the Moon in orbit around the Earth, and holds galaxies together. The gravitational constant, G, is the same everywhere.
4. What is the difference between G (universal gravitational constant) and g (acceleration due to gravity)?
The key difference lies in their constancy and what they measure:
- G (Universal Gravitational Constant): It is a fundamental constant of nature with a fixed value of 6.674 × 10⁻¹¹ N·m²/kg². Its value is the same everywhere in the universe.
- g (Acceleration due to Gravity): It is the acceleration experienced by an object due to the gravitational pull of a large body like a planet. Its value is not constant and changes depending on the planet's mass and radius. On Earth's surface, its average value is approximately 9.8 m/s².
5. What are some important applications of the Universal Law of Gravitation?
The Universal Law of Gravitation has many crucial applications and helps explain several natural phenomena, including:
- The force that binds us and all objects to the Earth.
- The motion of the Moon orbiting the Earth and planets orbiting the Sun.
- The formation of tides in the oceans, caused by the gravitational pull of the Moon and the Sun.
- Predicting the orbits of artificial satellites and space probes.
6. What is the difference between the terms 'gravitation' and 'gravity'?
While often used interchangeably, there is a subtle difference:
- Gravitation refers to the universal force of attraction between any two objects with mass.
- Gravity specifically refers to the gravitational force exerted by a very large celestial body (like a planet or a star) on an object near its surface. For example, the force pulling a ball back to the ground is called Earth's gravity.
7. If every object pulls on every other object, why don't we feel the gravitational pull from people or things around us?
We don't feel these forces because the gravitational force is extremely weak unless at least one of the objects has an immense mass. The universal gravitational constant (G) is a very small number. While a gravitational force technically exists between you and a book, your masses are so small that the resulting force is negligible and easily overcome by other forces like friction.
8. Who formulated the Universal Law of Gravitation?
The Universal Law of Gravitation was formulated by Sir Isaac Newton. He published this groundbreaking law in his book 'Philosophiæ Naturalis Principia Mathematica' in 1687. It was a revolutionary idea that connected the force causing objects to fall on Earth with the celestial forces governing planetary motion.
9. Does Newton's Universal Law of Gravitation have any limitations?
Yes, while Newton's law is highly accurate for most practical purposes, it has limitations in extreme conditions. It becomes less accurate in very strong gravitational fields (like near a black hole) or for objects moving at speeds close to the speed of light. In these scenarios, Albert Einstein's Theory of General Relativity provides a more precise description of gravity.
