Introduction and Universal Law of Gravitation

Gravitation

We observe that objects fall to the ground when dropped from a height. This familiar phenomenon is due to the force of gravitation, the fundamental force of attraction between any two objects possessing mass.

It is this same force that keeps us anchored to the Earth's surface, makes rain fall downwards, causes rivers to flow towards the sea, and is responsible for the motion of celestial bodies.

Before Newton, it was commonly believed that different laws governed the motion of objects on Earth and the motion of celestial bodies in the heavens. However, Isaac Newton, inspired by the famous story of a falling apple (which may or may not be literally true, but illustrates the concept), realised that the same force responsible for the apple falling to the ground might also be responsible for keeping the Moon in orbit around the Earth, and the planets in orbit around the Sun.

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Universal Law Of Gravitation

Based on his insights and the meticulous observations of planetary motion by astronomers like Tycho Brahe and the analysis by Johannes Kepler, Newton formulated the Universal Law of Gravitation. This law describes the attractive force between any two objects with mass, anywhere in the universe.

The law states that every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

We will delve into the detailed mathematical formulation and implications of this law later in this topic.

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Importance Of The Universal Law Of Gravitation

Newton's Universal Law of Gravitation was a monumental achievement in physics. Its importance lies in its ability to explain a vast range of phenomena with a single, simple principle. Its significance includes:

1. Explaining Terrestrial Phenomena: It explains why objects fall towards the Earth, the variation of weight with altitude, and the effect of gravity on fluids.
2. Explaining Celestial Phenomena: It successfully explains the motion of the Moon around the Earth, the planets around the Sun, and the orbits of comets and asteroids. It provided a physical basis for Kepler's Laws of planetary motion.
3. Predicting Astronomical Events: The law allowed for accurate predictions of planetary positions and led to the discovery of new planets (like Neptune, based on perturbations in Uranus's orbit).
4. Unifying Earthly and Heavenly Mechanics: It demonstrated that the same fundamental laws of physics apply everywhere in the universe, unifying the mechanics of terrestrial objects and celestial bodies.
5. Foundation for Future Physics: It laid the groundwork for numerous developments in physics and astronomy and remains essential for understanding satellite orbits, space travel, and the structure of galaxies.

This law is truly 'universal' because it applies to all objects, regardless of their size, composition, or location in the cosmos.

Introduction

The phenomenon of gravitation is one of the most pervasive forces in nature, governing the structure and motion of everything from the smallest particles to the largest galaxies. On Earth, gravity is the force that gives objects weight, causes them to fall when unsupported, and influences the flow of water and air.

Historically, the understanding of gravity evolved over centuries. Ancient civilisations observed the predictable motion of stars and planets but often attributed them to divine intervention or complex celestial spheres. Galileo Galilei made significant contributions through his experiments, demonstrating that objects of different masses fall at the same rate in the absence of air resistance, challenging the Aristotelian view that heavier objects fall faster.

The pivotal moment came with Sir Isaac Newton in the late 17th century. Legend has it that observing an apple fall from a tree prompted him to consider if the same force pulling the apple down also pulls on the Moon, keeping it in orbit around the Earth. Prior to this, earthly physics (mechanics) and celestial physics (astronomy) were often treated as separate domains with different rules.

Newton's genius was to propose a single, unifying force – gravitation – that acts between any two objects with mass, whether they are an apple and the Earth, or the Earth and the Moon, or the Sun and a planet. This bold hypothesis led to the formulation of his Universal Law of Gravitation, revolutionising our understanding of the universe.

This law successfully explained why planets orbit the Sun in elliptical paths (as described by Kepler's Laws), why the Moon influences tides, and why the acceleration due to gravity varies slightly across the Earth's surface and with altitude. It became the first fundamental force to be described mathematically, paving the way for understanding other forces in nature.

In this section, we will delve deeper into the precise statement and mathematical form of this law and explore the significance of the gravitational constant.

Universal Law Of Gravitation

Newton's Universal Law of Gravitation quantifies the attractive force between any two point masses. It provides a mathematical relationship that allows us to calculate the magnitude and direction of this force.

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Statement of the Law

As stated before, the law says:

Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

The direction of this force is along the line joining the two particles.

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Mathematical Formulation

Consider two objects (approximated as point masses) with masses $m_1$ and $m_2$. Let the distance between their centres be $r$. According to the law, the magnitude of the gravitational force ($F$) acting between them is:

$ F \propto m_1 m_2 $ (directly proportional to the product of masses)

$ F \propto \frac{1}{r^2} $ (inversely proportional to the square of the distance)

Combining these two proportionalities, we get:

$ F \propto \frac{m_1 m_2}{r^2} $

To convert this proportionality into an equation, we introduce a constant of proportionality, which is the Universal Gravitational Constant, denoted by $G$.

The formula for the magnitude of the gravitational force is:

$ F = G \frac{m_1 m_2}{r^2} $

where:

• $F$ is the magnitude of the gravitational force between the two objects (in Newtons, N).
• $G$ is the Universal Gravitational Constant (discussed in detail later).
• $m_1$ and $m_2$ are the masses of the two objects (in kilograms, kg).
• $r$ is the distance between the centres of the two objects (in metres, m).

The force is always attractive, meaning $m_1$ pulls on $m_2$, and $m_2$ pulls on $m_1$. By Newton's Third Law of Motion, the force exerted by $m_1$ on $m_2$ is equal in magnitude and opposite in direction to the force exerted by $m_2$ on $m_1$.

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Vector Form of the Law

Gravitational force is a vector quantity. The vector form of the Universal Law of Gravitation specifies both the magnitude and direction of the force.

Let $\vec{r}_{12}$ be the position vector from $m_1$ to $m_2$, and $\hat{r}_{12}$ be the unit vector in the direction from $m_1$ to $m_2$. Similarly, let $\vec{r}_{21}$ be the position vector from $m_2$ to $m_1$, and $\hat{r}_{21}$ be the unit vector in the direction from $m_2$ to $m_1$. Note that $\vec{r}_{12} = -\vec{r}_{21}$ and $\hat{r}_{12} = -\hat{r}_{21}$. The magnitude of both position vectors is $r = |\vec{r}_{12}| = |\vec{r}_{21}|$.

The force exerted on $m_2$ by $m_1$, denoted by $\vec{F}_{21}$, is directed towards $m_1$. The direction from $m_2$ to $m_1$ is given by the unit vector $\hat{r}_{21}$ (or $-\hat{r}_{12}$). Since the force is attractive, $\vec{F}_{21}$ is in the direction of $\hat{r}_{21}$.

$ \vec{F}_{21} = G \frac{m_1 m_2}{r^2} \hat{r}_{21} $

Alternatively, since $\hat{r}_{21} = -\hat{r}_{12}$, we can write:

$ \vec{F}_{21} = -G \frac{m_1 m_2}{r^2} \hat{r}_{12} $

This shows that the force on $m_2$ is towards $m_1$, opposing the direction of $\hat{r}_{12}$.

Similarly, the force exerted on $m_1$ by $m_2$, denoted by $\vec{F}_{12}$, is directed towards $m_2$. The direction from $m_1$ to $m_2$ is given by the unit vector $\hat{r}_{12}$.

$ \vec{F}_{12} = G \frac{m_1 m_2}{r^2} \hat{r}_{12} $

From these vector expressions, it is clear that $\vec{F}_{21} = -\vec{F}_{12}$, confirming Newton's Third Law for gravitational forces.

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Principle of Superposition

If there are multiple objects interacting gravitationally, the net gravitational force on any one object is the vector sum of the individual gravitational forces exerted on it by all other objects. This is known as the Principle of Superposition.

If we have a system of $N$ objects with masses $m_1, m_2, ..., m_N$ and positions $\vec{r}_1, \vec{r}_2, ..., \vec{r}_N$, the net force $\vec{F}_i$ on the $i$-th object with mass $m_i$ is the vector sum of the forces exerted by all other objects ($j \ne i$):

$ \vec{F}_i = \sum_{j=1, j \ne i}^{N} \vec{F}_{ij} $

where $\vec{F}_{ij}$ is the force exerted on $m_i$ by $m_j$. Using the vector form of the law:

$ \vec{F}_i = \sum_{j=1, j \ne i}^{N} G \frac{m_i m_j}{|\vec{r}_i - \vec{r}_j|^2} \hat{r}_{ij} $

where $\hat{r}_{ij}$ is the unit vector from $m_j$ to $m_i$, i.e., $\hat{r}_{ij} = \frac{\vec{r}_i - \vec{r}_j}{|\vec{r}_i - \vec{r}_j|}$.

Answer:

The Gravitational Constant

The equation for Newton's Universal Law of Gravitation, $F = G \frac{m_1 m_2}{r^2}$, includes a proportionality constant $G$. This constant is known as the Universal Gravitational Constant. It is called 'universal' because it is believed to have the same value everywhere in the universe.

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What does G Represent?

$G$ represents the strength of the gravitational force. Its numerical value tells us how strong the gravitational attraction is between objects of given masses separated by a given distance. A larger value of $G$ would mean stronger gravitational forces, and a smaller value would mean weaker forces.

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Value and Units of G

The value of $G$ is extremely small, which is why gravitational forces are noticeable only when at least one of the interacting objects has a very large mass, like a planet or a star. The precise value of $G$ is difficult to measure accurately because the gravitational force between laboratory-sized objects is very weak.

The accepted value of the Universal Gravitational Constant, as recommended by CODATA (Committee on Data for Science and Technology), is approximately:

$ G = 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $

The units of $G$ can be determined from the gravitational force formula $F = G \frac{m_1 m_2}{r^2}$. Rearranging for $G$: $G = \frac{F r^2}{m_1 m_2}$. Substituting the SI units (N for force, m for distance, kg for mass):

$ \text{Units of } G = \frac{\text{N} \times \text{m}^2}{\text{kg} \times \text{kg}} = \frac{\text{N m}^2}{\text{kg}^2} $

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Measurement of G (Cavendish Experiment)

The value of $G$ was first experimentally determined by the English scientist Henry Cavendish in 1798, about 71 years after Newton's death. His experiment, known as the Cavendish Experiment, used a sensitive torsion balance to measure the very weak gravitational attraction between small lead spheres and large lead spheres.

(Image Placeholder: A diagram showing a horizontal rod suspended by a thin wire, with two small spheres attached at the ends of the rod. Two larger spheres are brought near the small spheres, showing the gravitational attraction causing the rod to twist, and indicating how the twist angle is measured.)

By measuring the small angle of twist in the suspension wire caused by the gravitational force between the masses, and knowing the torsion constant of the wire, Cavendish was able to calculate the gravitational force. With known masses and distance, he could then determine the value of $G$. This experiment was not only important for finding $G$ but also, indirectly, allowed for the calculation of the Earth's mass and average density for the first time, as the gravitational force exerted by the Earth on an object of known mass could be compared to the measured force between the spheres.

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Distinction between $G$ and $g$

It is important not to confuse the Universal Gravitational Constant ($G$) with the acceleration due to gravity ($g$).

Feature	Universal Gravitational Constant (G)	Acceleration due to Gravity (g)
What it is	A fundamental constant representing the strength of gravitational interaction between any two masses.	The acceleration experienced by an object falling freely near a massive body (like Earth) due to that body's gravitational pull.
Value	$6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2$ (A very small, constant value).	Approximately $9.8 \, \text{m/s}^2$ on the Earth's surface (varies with location, altitude, etc.).
Units	$\text{N m}^2/\text{kg}^2$	$\text{m/s}^2$ (which are equivalent to N/kg).
Dependence	A universal constant; does NOT depend on the masses, distance, or medium between the objects.	Depends on the mass and radius of the celestial body causing the gravity, and the distance from its centre. Also affected by local factors (altitude, latitude, density of crust).
Vector/Scalar	Scalar constant (part of a vector force formula).	Vector quantity (directed towards the centre of the massive body).

The acceleration due to gravity 'g' on the surface of a planet of mass $M$ and radius $R$ is given by $g = G \frac{M}{R^2}$. This equation highlights how $g$ is derived from the universal constant $G$ and the properties of the specific planet.