Correct Answer - Option 4 : None of the above
The correct answer is option 4) i.e. None of the above
CONCEPT:
-
Acceleration due to gravity on the surface of Earth of mass M and radius Re is denoted by g.
- It has an approximated uniform value of 9.8 m/s2 on the surface of Earth.
The acceleration due to gravity at a depth 'd' below the surface of Earth is given by:
\(⇒ g' = g(1- \frac{d}{R_e})\)
- The acceleration due to gravity at a height 'h' above the surface of Earth is given by:
\(⇒ g'' = g(1+ \frac{h}{R_e})^{-2}\)
\(⇒ g'' = g(1- \frac{2h}{R_e})\) for h << Re
EXPLANATION:
- As we go deeper into the earth's surface, d increases. From the equation, \(g' = g(1- \frac{d}{R_e})\) it can be inferred that g' decreases on increasing d. Thus, acceleration due to gravity decreases with depth.
- At heights above the surface of the earth i.e. r > R, it can be inferred from the equation, \(g'' = g(1+ \frac{h}{R_e})^{-2}\) that g'' decreases on increasing r beyond R. Thus, acceleration due to gravity decreases with altitude.
- From the Law of Universal Gravitation, the gravitational force acting on an object of mass m placed on the surface of Earth is:
\(F = \frac{GMm}{R^2}\) ----(1)
Where R is the radius of the earth.
From newtons' second law of motion, we know F = ma = mg ----(2)
Therefore, from (1) and (2),
\(g =\frac{GM}{R^2}\)
- So, acceleration due to gravity is dependent on the mass of the Earth.
-
Therefore, none of the given options is correct.
