Problem 21 - Master Resource Book in JEE Main Physics

Question

The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity \(\quad\) [NCERT Exemplar] (a) will be directed towards the centre but not the same everywhere (b) will have the same value everywhere but not directed towards the centre (c) will be same everywhere in magnitude directed towards the centre (d) cannot be zero at any point

Step-by-Step Solution

Answer

Option (a) is correct.

Step 1: Understanding Gravity on Earth

On Earth, the force of gravity acts as a central force, which means it is directed towards the center of the Earth. This is due to the Earth having an approximate spherical shape.

Step 2: Analyzing Non-Uniform Density

If the Earth's interior is not of uniform density, gravitational field lines can be deflected, and gravitational acceleration may vary across different points on the Earth's surface.

Step 3: Evaluating the Options

Given that density variations can affect the gravitational field, let's examine each option: (a) Directed towards the centre but not the same everywhere - consistent with varying gravitational magnitudes due to non-uniform density. (b) Same value everywhere - inconsistent with non-uniform density. (c) Same in magnitude towards the centre - also inconsistent with density variations. (d) Cannot be zero - consistent, as gravity doesn't become zero just due to density variations.

Step 4: Selecting the Correct Option

Option (a) will be directed towards the centre but not the same everywhere, is accurate because the non-uniform density causes variations in gravitational acceleration across different points on the Earth’s surface.

Key Concepts

Uniform Density

Uniform density is a concept where a material, object, or planet has the same density throughout its entire structure. On Earth, if it had uniform density, its mass would be evenly distributed. This would mean that the gravitational acceleration, which is the force of gravity acting per unit mass, is constant at any given distance from the center of the Earth.

Most planets, including Earth, do not have uniform density because their interiors are composed of different materials with varying densities, such as the core, mantle, and crust. These different layers affect how mass is distributed inside the Earth. As a result, the gravitational force varies slightly at different points on the surface.

• Consistent Force: With uniform density, gravitational acceleration would be the same everywhere on Earth's surface.
• Real Earth: Earth's non-uniform density means this consistency doesn't occur; instead, there are slight variations.

Understanding density distribution is crucial when predicting precise gravitational forces and explaining the specific paths of satellites orbiting the earth.

Central Force

The concept of a central force is where the force acts along the line connecting the center of an object and a point outside the object, and in the case of gravity, it acts towards the center of mass. In simple terms, any force that pulls objects towards a central point is a central force.

• On Earth, gravity is a central force, meaning it pulls you towards the center of the Earth, regardless of where you are on its surface.
• This force is what keeps the atmosphere in place and causes objects to fall towards the ground when released.

The gravitational pull, when seen as a central force, not only keeps objects bound to the Earth but also governs the motion of the planets around the sun, as each planet experiences a central force due to the sun's gravitational pull.

In celestial mechanics, understanding the nature of central forces is key to exploring and predicting planetary orbits and other large-scale motions in the universe. Central forces help us understand why objects in space move the way they do, leading us to new insights into the dynamic universe.

Gravitational Field Lines

Gravitational field lines are an imaginary representation used to visualize the gravitational field around a mass, like the Earth. These lines illustrate the direction and strength of the gravitational pull exerted by a mass.

• Direction: They always point towards the center of the mass causing the gravitational field, as they represent the path a free particle would take under the influence of gravity.
• Density: The closer the lines are to each other, the stronger the gravitational field is at that point.

In a scenario where Earth's density is non-uniform, variations in these lines would occur. This means they could appear denser or more dispersed depending on the density of the surrounding matter. When density changes in the Earth's interior, these lines can curve or bend slightly, indicating variations in gravitational acceleration at different points on the Earth's surface.

By studying gravitational field lines, scientists can infer the structure and behavior of Earth's interior and understand how mass is distributed within other celestial bodies too. They are a vital tool in the fields of geophysics and astrophysics, helping to map out gravitational interactions in space.