This chapter lays the foundation for understanding the fundamental force that governs the motion of celestial objects and everyday interactions with gravity. We’ll explore Newton’s law of universal gravitation, delve into the vector nature of gravitational force, and understand the principle of superposition for combining gravitational interactions.

## 1. Universal Law of Gravitation

Sir Isaac Newton revolutionized our understanding of gravity with his universal law of gravitation. This law states:

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

Mathematically, it can be expressed as:

**F = G * (m1 * m2) / r^2**

where:

- F is the gravitational force between the two particles (measured in Newtons)
- G is the universal gravitational constant (approximately 6.6743 x 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two particles (measured in kilograms)
- r is the distance between the centers of the particles (measured in meters)

**Key Points:**

- Gravity is always attractive, meaning it pulls objects together.
- The strength of the force increases with the product of the masses and decreases with the square of the distance.
- The universal gravitational constant, G, is a constant value for all interactions in the universe.

## 2. Gravitational Force in Vector Form

While the scalar form of the law provides the magnitude of the force, the vector form specifies its direction. The gravitational force acts along the line joining the centers of the two masses. We can express it as:

**F = G * (m1 * m2) * ((r1 – r2) / |r1 – r2|^2)**

where:

- F is the gravitational force vector (measured in Newtons)
- G is the universal gravitational constant
- m1 and m2 are the masses of the two particles
- r1 and r2 are the position vectors of the centers of mass 1 and mass 2, respectively (measured in meters)

**Table 1: Comparison of Scalar and Vector Forms**

Feature | Scalar Form | Vector Form |
---|---|---|

Magnitude | F | |

Direction | Not specified | Along the line joining centers |

Understanding the vector nature of gravity becomes crucial when dealing with situations involving multiple interacting masses.

## 3. Principle of Superposition

The principle of superposition states that the net gravitational force acting on a particle due to several other particles is the vector sum of the individual gravitational forces exerted by each particle.

In simpler terms, for a particle interacting with multiple other masses, the total force it experiences is the sum of the individual forces exerted by each mass, taking into account their directions.

Mathematically, the net force can be found using vector addition:

**F_net = Σ (G * (m_i * m) / r_i^2) * ((r_i – r) / |r_i – r|^2)**

where:

- F_net is the net gravitational force acting on the particle (measured in Newtons)
- G is the universal gravitational constant
- m is the mass of the particle (measured in kilograms)
- m_i represents the mass of each of the other interacting particles (measured in kilograms)
- r_i represents the position vector of the center of mass i (measured in meters)
- r represents the position vector of the particle experiencing the net force (measured in meters)

## 4. Gravitational Field: Meaning

The gravitational field at a point in space is defined as the gravitational force per unit mass experienced by a small test mass placed at that point. It’s a vector field, meaning it has both magnitude and direction. The direction of the gravitational field coincides with the direction of the force a test mass would experience.

**Key Points:**

- The gravitational field is independent of the test mass’s mass.
- A stronger gravitational field signifies a stronger force experienced by a test mass.

## 5. Gravitational Field of Different Objects

The gravitational field created by various objects can be calculated using the following formula:

**g = G * (M / r^2)**

where:

- g is the gravitational field strength (measured in N/kg or m/s²)
- G is the universal gravitational constant
- M is the mass of the object creating the field (measured in kilograms)
- r is the distance from the center of mass of the object to the point where the field is being measured (measured in meters)

The following sections will explore the gravitational field due to various objects:

### 5.1 Point Mass

A point mass is an object whose entire mass is concentrated at a single point. The gravitational field due to a point mass is:

**g = G * (M / r^2)**

This is the most basic form of the formula, applicable to any object considered a point mass for a specific calculation.

### 5.2 System of Point Masses

For a system of point masses, the principle of superposition applies. The net gravitational field at a point is the vector sum of the individual fields created by each point mass.

### 5.3 Linear Rod

**On the axis of a finite rod:**

The gravitational field due to a finite rod on its axis can be calculated using integration. However, for JEE Main, it’s sufficient to understand that the field isn’t uniform and increases as you move closer to the ends of the rod.

**Perpendicular line passing through one end of the rod:**

The field along this line increases initially and then reaches a constant value due to the cancellation of effects from opposite sides of the rod beyond a certain point.

### 5.4 Semi-Infinite Rod

The gravitational field due to a semi-infinite rod on its axis follows the same principle as a finite rod but only for the half-length portion.

### 5.5 Infinite Rod

The gravitational field due to an infinite rod at any point is constant along its axis and has a magnitude of:

**g = 2Gλ**

where λ (lambda) is the linear mass density of the rod (mass per unit length).

### 5.6 Ring

**On the axis of a ring:**

The gravitational field of a ring on its axis can be calculated using integration, but a qualitative understanding suffices for JEE Main. The field varies depending on the distance from the center of the ring.

**At the center of the ring:**

The gravitational field due to a ring at its center is zero due to symmetry.

### 5.7 Solid Sphere

The gravitational field due to a solid sphere:

- Inside the sphere: The field increases linearly with distance from the center.
- Outside the sphere: The field is identical to that of a point mass with all its mass concentrated at the center. This is due to the Gauss’s law application in spherical symmetry.

### 5.8 Spherical Shell

The gravitational field due to a spherical shell:

- Inside the shell: The field is zero due to the cancellation of effects from opposite sides of the shell.
- Outside the shell: The field is again identical to that of a point mass with all its mass concentrated at the center.

Remember, for JEE Main, a qualitative understanding of the behavior of the field for these objects is sufficient. Focus on practicing problems involving calculations for simpler cases like point masses and spheres.

## 6. Variation of Acceleration Due to Gravity (g) with Earth’s Radius

The acceleration due to gravity (g) experienced by an object isn’t constant throughout Earth. It weakens with increasing distance from Earth’s center and strengthens with decreasing distance. Here’s how it varies:

**6.1. Variation with Height Above Earth’s Surface:**

As you move farther away from Earth’s surface (increasing height), the acceleration due to gravity weakens. This is because the distance (r) in the formula **g = G * M / r^2** increases, resulting in a smaller g value (G and M being constants for Earth).

**6**.**2. Variation with Depth Below Earth’s Surface:**

The situation is a bit more nuanced when considering depth below the surface. Here’s the breakdown:

**Initially (shallow depths):**Similar to increasing height, g weakens slightly as you descend because you’re getting farther from the total mass of Earth.**As you go deeper:**The mass above you decreases, but the mass below you increases, and its gravitational pull becomes more significant. This leads to a gradual increase in g until you reach the center.

The change in g with depth or height for practical applications near Earth’s surface is minimal. For most JEE Main problems, you can assume a constant value for g (around 9.8 m/s²). However, it’s crucial to understand the underlying concept for deeper understanding.

## 7. Influence of Earth’s Rotation and Shape/Size on Gravity (g)

While Earth’s rotation and its shape/size do affect the experienced acceleration due to gravity (g) slightly, these effects are usually negligible for most JEE Main problems. Here’s a breakdown:

**7**.**1. Due to Rotation of Earth:**

Earth isn’t a perfect sphere; it bulges slightly at the equator due to its rotation. This creates a centrifugal force (outward force) that acts opposite to gravity.

**Effect:**The centrifugal force is strongest at the equator and weakens towards the poles. This counteracts gravity slightly, leading to a**minor decrease**in g at the equator compared to the poles.

**7.2. Due to Shape and Size of Earth:**

Earth’s oblate spheroid shape (bulging equator) and its overall size play a role in determining g.

**Effect:**Due to the bulge, an object at the equator is slightly farther from Earth’s center compared to an object at the poles. This, again, leads to a**minor decrease**in g at the equator.

**7.3** **Combined Effect:**

Both the rotation and the shape contribute to a **net decrease** in g at the equator compared to the poles. However, this difference is very small. The value of g at the equator is around 9.78 m/s², while at the poles, it’s about 9.83 m/s².

## 8. Gravitational Potential Energy

Gravitational potential energy (PE) is a form of energy possessed by an object due to its position in a gravitational field. It represents the potential an object has to convert that energy into kinetic energy as it moves within the field.

Imagine lifting a book off the ground. You’re working against the gravitational pull of Earth. The higher you lift the book, the more work you do, and the more potential energy the book gains. If you let go, the gravitational force pulls the book down, converting its gravitational potential energy into kinetic energy of motion.

- Gravitational PE is a scalar quantity, meaning it has only magnitude and no direction.
- It depends on the object’s mass (m) and its position (h) relative to a chosen reference point (often considered Earth’s surface).
- The PE is zero at the reference point and increases as the object moves farther away against the gravitational pull.

**9**. **Types of Gravitational PE**

There are two main ways to categorize gravitational PE:

**9**.**1. Interaction PE:**

This refers to the gravitational potential energy an object possesses due to its interaction with another massive object, typically Earth. It’s the type of PE discussed previously, where lifting an object increases its PE relative to Earth’s surface.

**9.2. Self PE (for extended objects):**

This concept applies to objects with significant size and mass distribution, like planets or stars. Due to the gravitational attraction between different parts of the object itself, it can possess some internal gravitational PE. However, calculating self PE is complex and often not required for JEE Main.

**Formula for Interaction PE:**

The gravitational PE (PE) of an object due to its interaction with another massive object is given by:

**PE = m * g * h**

where:

- m is the mass of the object (kg)
- g is the acceleration due to gravity (usually taken as a constant value, around 9.8 m/s² for JEE Main)
- h is the vertical height of the object above the reference point (meters)

**Understanding the Formula:**

- The product (m * g) represents the weight of the object, the force due to gravity acting on it.
- Multiplying weight by height (h) gives the work done against gravity to lift the object to that position. This work done translates to the object’s gained gravitational PE.

**10.** **Gravitational Potential**

Gravitational potential (V) is closely related to gravitational potential energy (PE) but offers a slightly different perspective. It describes the work done per unit mass required to move an object from a reference point (often infinity) to a specific point within a gravitational field.

**Meaning:**

Imagine a point in space within Earth’s gravitational field. The gravitational potential at that point tells you how much work you’d need to do (per unit mass) to bring an object from a very far distance (considered infinity for simplicity) to that specific point, entirely against the gravitational pull.

**Key Points:**

- Gravitational potential is a scalar quantity, meaning it has only magnitude and no direction.
- A lower potential signifies less work needed to bring a mass from infinity.
- The reference point for potential is arbitrary; however, choosing infinity makes calculations often more manageable.

**Potential of Different Objects:**

The gravitational potential due to various objects can be calculated using the following general formula:

**V = – G * (M / r)**

where:

- V is the gravitational potential (J/kg)
- G is the universal gravitational constant
- M is the mass of the object creating the field (kg)
- r is the distance from the center of mass of the object to the point where the potential is being measured (meters)

The negative sign indicates that work needs to be done against the attractive gravitational force.

Let’s explore the potential for some specific objects:

**10**.**1. Point Mass:**

For a point mass, the formula applies directly:

**V = – G * (M / r)**

**10.2. Solid Sphere:**

The gravitational potential inside a uniform solid sphere **increases linearly with distance from the center**. This is because as you move inwards, the amount of mass attracting you increases, requiring more work per unit mass to overcome gravity.

**10.3. Spherical Shell:**

The interesting aspect of a spherical shell is that the gravitational potential due to a uniform spherical shell at any point **outside the shell** is the same as that of a point mass with all its mass concentrated at the center. This applies due to the symmetrical cancellation of gravitational effects from opposite sides of the shell. However, the potential **inside the shell is zero**.