Question

The variation of acceleration due to gravity $g$ with distance $d$ from centre of the earth is best represented by (R = Earth?s radius) :

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The Correct Option is D

To solve this problem, we need to understand how acceleration due to gravity \(g\) varies with distance \(d\) from the center of the Earth. 

The acceleration due to gravity at a distance \(d\) from the center of the Earth can be expressed differently depending on whether \(d\) is inside or outside the Earth's surface.

1. When \(\)

The acceleration due to gravity is directly proportional to the distance from the center, i.e., \(g \propto d\). The formula is \(g = \frac{G M d}{R^3}\), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the Earth.

1. When \(d \geq R\) (outside or on the surface of the Earth):

The acceleration due to gravity is inversely proportional to the square of the distance from the center, i.e., \(g \propto \frac{1}{d^2}\). The formula is \(g = \frac{G M}{d^2}\).

Given these relationships, the plot of \(g\) versus \(d\) can be visualized as follows:

1. The graph is initially linear, increasing from zero as \(d\) rises from the center towards the surface (since \(g \propto d\)).
2. Beyond the surface of the Earth, the graph becomes a curve that decreases as \(d\) increases (reflecting the inverse square relationship).

This behavior is reflected in the correct image:

This plot correctly shows a linear increase of \(g\) with \(d\) inside the Earth and a sharp decrease as \(g\) becomes inversely proportional to \(d^2\) outside the Earth.