Question

In the reported figure of the earth, the value of acceleration due to gravity is the same at points A and C but it is smaller than that of its value at point B (surface of the earth). The value of OA: AB will be x: 5. Then find the value of x.

Answer 1

Hint:Before solving this problem let’s understand acceleration due to gravity. Acceleration due to gravity is defined as the acceleration gained by an object due to gravitational force and its S.I. unit is \[m{s^{ - 2}}\]. It is a vector quantity where it has both magnitude and direction.

Formula Used:
The acceleration due to gravity at depth d is,
\[g = g\left( {1 - \dfrac{d}{{{R_e}}}} \right)\]
Where, d is the depth of the earth’s surface and \[{R_e}\] is radius of earth.

Complete step by step solution:

Image: The figure of Earth.

Consider the earth as shown in the figure where the three points A, B, and C are given. B is a point on the surface of the earth, C is placed 3200 km above the earth and A is a point below the earth’s surface (depth). They have said that the acceleration due to gravity at A is equal to the acceleration due to gravity at C. That is,
\[{g_A} = {g_{_C}}\]

At depth from the earth’s surface, the value of acceleration due to gravity at point A is,
\[{g_A} = g\left( {1 - \dfrac{d}{{{R_e}}}} \right)\]
The value of acceleration due to gravity at point C is,
\[{g_C} = \dfrac{g}{{{{\left( {1 + \dfrac{h}{{{R_e}}}} \right)}^2}}}\]
Here, h is the height from earth’s surface, that is, \[h = 3200\,km\] and \[{R_e} = 6400\,km\].

Then the above equation will become,
\[{g_C} = \dfrac{g}{{{{\left( {1 + \dfrac{{3200}}{{6400}}} \right)}^2}}}\]
\[\Rightarrow {g_C} = \dfrac{g}{{{{\left( {1 + \dfrac{1}{2}} \right)}^2}}}\]
\[\Rightarrow {g_C} = \dfrac{g}{{{{\left( {\dfrac{3}{2}} \right)}^2}}}\]
\[\Rightarrow {g_C} = \dfrac{{4g}}{9}\]
Since, \[{g_A} = {g_{_C}}\]
\[g\left( {1 - \dfrac{d}{{{R_e}}}} \right) = \dfrac{{4g}}{9}\]
\[\Rightarrow \left( {1 - \dfrac{4}{9}} \right) = \dfrac{d}{{{R_e}}}\]
\[\Rightarrow d = \dfrac{{5{R_e}}}{9}\]

From the figure,
\[d = AB\]
\[\Rightarrow AB = \dfrac{{5{R_e}}}{9}\]
Then, \[OA = R - AB\]
\[OA = R - \dfrac{{5{R_e}}}{9}\]
\[\Rightarrow OA = \dfrac{{4{R_e}}}{9}\]
Given that, OA: AB will be x: 5 that is,
\[\dfrac{{OA}}{{AB}} = \dfrac{x}{y}\]
\[\Rightarrow \dfrac{{\dfrac{{4{R_e}}}{9}}}{{\dfrac{{5{R_e}}}{9}}} = \dfrac{x}{y}\]
\[ \therefore \dfrac{x}{y} = \dfrac{4}{5}\]

Therefore, the value of x is 4.

Note:Remember to keep in mind the locations of all the provided points with respect to the earth’s surface as shown in the diagram. Then it will be easier to find the value of x.