The value of the acceleration due to gravity is at a height ( = radius of the earth) from the surface of the earth. It is again equal to at a depth below the surface of the earth. The ratio equals :
Step 1: Given data
Acceleration due to gravity at height is .
Here,
Acceleration due to gravity at depth is also .
Here, is the height above the earth's surface and is the depth from the earth's surface.
Step 2: Formula used:
Where, is the acceleration due to gravity,
is gravitational constant,
is the mass of Earth and is the radius of the earth.
Density is defined as mass per unit volume and is given by,
The volume of a sphere is given as,
Step 3: Calculating the acceleration due to gravity at a height h
As we know that
Therefore, acceleration due to gravity at height h will be,
Substitute the value for height as in the above formula,
Step 4: Calculating the mass of the earth at a depth d.
While calculating the acceleration due to gravity at a depth "d" we need to consider the mass of the earth only up depth "d" from the centre of the earth because the mass that is participating in the acceleration due to gravity is only up to a depth “d” as shown by the red color in the figure.
Now we need to calculate the mass of the earth at a depth d.
The density of the earth can be calculated by,
Where volume
Putting the value of volume we get,
Now, since the density will remain the same even at a depth so we can calculate the mass of the earth at depth "d",
Where, is the mass of earth at a depth “d”.
Step 5: Calculating the acceleration due to gravity at a depth "d".
Putting the value of in the above equation, we get,
Step 6: Calculating the ratio of
Now in the question, it is given that the value of acceleration due to gravity is the same both at height “h” and depth “d”.
Therefore, (given in question).
Now on dividing equations (1) and (2), we get,
Therefore the correct answer is option (C).