The Universal Law of Gravitation is one of the most fundamental principles in physics, formulated by Sir Isaac Newton in 1687. This law explains how every object in the universe attracts every other object with a force that depends on their masses and the distance between them.
For Class 9 students, understanding the derivation of the Universal Law of Gravitation is crucial for building a strong foundation in physics. In this topic, we will explore Newton’s Universal Law of Gravitation, its derivation, formula, and real-world applications in an easy-to-understand manner.
What is Newton’s Universal Law of Gravitation?
Newton’s Universal Law of Gravitation states that:
“Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”
Mathematical Expression of the Law
Mathematically, the gravitational force F between two objects of masses m₁ and m₂, separated by a distance r, is given by:
Where:
- F = Gravitational force between two objects (in Newtons, N)
- G = Universal Gravitational Constant ( $6.674 times 10^{-11} , text^2/text^2$ )
- m₁, m₂ = Masses of the two objects (in kg)
- r = Distance between the centers of the two objects (in meters)
This equation forms the basis of gravitational interactions in the universe, from the motion of planets to the falling of objects on Earth.
Derivation of Universal Law of Gravitation
Newton derived the Universal Law of Gravitation based on Kepler’s laws of planetary motion and the observations of Galileo. The derivation is based on the assumption that gravitational force acts as a central force.
Step 1: Understanding the Proportional Relationships
(i) Force is directly proportional to the product of masses
From observations, Newton concluded that larger masses exert a stronger gravitational pull. Thus, the force F is directly proportional to the product of the two masses:
(ii) Force is inversely proportional to the square of the distance
Newton also observed that as the distance between two objects increases, the gravitational force decreases. More specifically, the force decreases as the square of the distance:
Step 2: Combining the Proportionalities
By combining the two proportional relationships, we get:
To remove the proportionality sign, we introduce a constant of proportionality, G (the Universal Gravitational Constant):
This is the Universal Law of Gravitation.
Understanding the Universal Gravitational Constant (G)
The constant G was first measured by Henry Cavendish in 1798 using the torsion balance experiment.
Value of G
Significance of G
- It determines the strength of gravitational attraction in the universe.
- It is extremely small, explaining why gravitational force is weak compared to other fundamental forces like electromagnetism.
Applications of Universal Law of Gravitation
The Universal Law of Gravitation has many important applications in physics and astronomy. Some of these include:
1. Motion of Planets and Satellites
- Newton’s law explains how planets orbit the Sun and how moons revolve around planets.
- It also governs the motion of artificial satellites like GPS satellites and space probes.
2. Free Fall of Objects on Earth
- Objects fall toward Earth due to gravitational attraction.
- The acceleration due to gravity g is derived from Newton’s law:
g = frac{G M}{R^2}
Where:
- M = Mass of Earth
- R = Radius of Earth
The standard value of g on Earth is 9.8 m/s².
3. Formation of Tides
- The gravitational pull of the Moon and Sun causes ocean tides on Earth.
- The Moon’s gravity pulls water toward it, creating high tides.
4. Black Holes and Gravitational Lensing
- Einstein’s General Relativity, which builds on Newton’s law, explains how black holes trap light due to extreme gravitational pull.
- Gravitational lensing occurs when massive objects like galaxies bend light from distant stars.
5. Weight Variation on Different Planets
- Since gravitational force depends on mass and radius, weight varies on different planets.
- A person’s weight on the Moon is 1/6th of their weight on Earth because the Moon’s gravity is weaker.
Importance of the Universal Law of Gravitation
1. Explains Large-Scale Interactions
- The law explains planetary motion, star formation, and galaxy interactions.
- It is used to calculate orbital velocities of celestial bodies.
2. Basis for Space Exploration
- NASA and space agencies use this law to design rockets, satellite trajectories, and space missions.
- It helps in launching and maneuvering spacecraft in zero gravity.
3. Governs Natural Phenomena
- Weather patterns, river flows, and even the human body’s weight distribution depend on gravity.
Limitations of Newton’s Universal Law of Gravitation
Although Newton’s law is highly accurate in most cases, it has some limitations:
1. Cannot Explain Relativity
- Newton’s law does not account for time dilation and space-time curvature.
- Einstein’s Theory of General Relativity (1915) provides a more precise model of gravity.
2. Fails at Atomic Scales
- Gravity is the weakest fundamental force and does not explain atomic interactions.
- Quantum mechanics governs subatomic ptopics, not Newton’s law.
3. Inaccurate for Extreme Gravity
- Near black holes and neutron stars, Newton’s equations break down.
- Einstein’s relativistic equations are needed for such cases.
Newton’s Universal Law of Gravitation is a cornerstone of classical physics, explaining how objects attract each other based on mass and distance.
Key Takeaways
- Formula: F = G frac{m_1 m_2}{r^2}
- Derivation: Based on direct proportionality to mass and inverse square proportionality to distance.
- Applications: Explains planetary motion, tides, free fall, space exploration, and more.
- Limitations: Does not explain relativity, quantum mechanics, or black hole physics.
Despite its limitations, Newton’s law remains one of the most important discoveries in physics, forming the basis for modern space science and engineering.