Question

As we go from equator to the poles, the value of $g$
A) Remains the same.
B) Decreases.
C) Increases.
D) Decreases upon an altitude of $45^\circ $.

Answer 1

Hint:The formula of force between the two planets can be used to calculate the correct answer for this problem. This formula is used to find the force between a planet and any other body. The variation of force with the distance between the two bodies can give us a clear indication about the value of g variation when we go from the equator to the poles.

Formula used: The formula of the acceleration due to gravity is given by $g = \dfrac{{GM}}{{{r^2}}}$ where G is gravitational constant M is the mass and r is the distance from the centre of earth.

Complete step by step answer:
The force between earth and another body can be given as,
$F = \dfrac{{G \cdot M \cdot {m_1}}}{{{r^2}}}$………eq. (1)
Where$M$ and ${m_1}$ are the masses of earth and a body and $r$is the distance from the center of the earth to the body.
We also know that the force applied on a body of mass ${m_1}$is given by,
$F = {m_1} \cdot g$………eq. (2)
Where F is the force on the body and g is the acceleration due to gravity.
So if we equate the force in equation (1) and equation (2) then we get,
$F = \dfrac{{G \cdot M \cdot {m_1}}}{{{r^2}}}$
Replace the value of F in equation (2) from equation (1).
$
  F = \dfrac{{G \cdot M \cdot {m_1}}}{{{r^2}}} \\
  {m_1} \cdot g = \dfrac{{G \cdot M \cdot {m_1}}}{{{r^2}}} \\
  g = \dfrac{{G \cdot M}}{{{r^2}}} \\
 $
Therefore, the acceleration due to gravity on the earth surface is given by $g = \dfrac{{GM}}{{{r^2}}}$ where G is gravitational constant, M is the mass of earth and r is the distance of any body from the centre of the earth. So as we move from equator to the poles the distance between the centre of earth and the body decreases and as g is inversely proportional to the distance for the centre of earth and which means that as we move from the equator to the poles the term $r$ will decrease and therefore the value g will increase. So the correct option for this problem is option C.

Note: The students should remember the formula of the force between two planets or force between a planet and anybody as this formula can help in solving these types of problems. The force on between two bodies of mass ${m_1}$ and ${m_2}$ are at a distance $r$ between two bodies is given by $F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ here it be observed that the value of force F will increase as the value of the radius r will decrease.