Question

The value of g at a particular point is 9.8. Suppose the earth suddenly shrinks uniformly to half its present size without losing any mass. The value of g at the same point (distance of the point from the centre of earth does not change) will now be
$\begin{align}
  & a)9.8 \\
 & b)4.9 \\
 & c)19.6 \\
 & d)39.2 \\
\end{align}$

Answer 1

Hint: Let us first find the gravitational acceleration in terms of known quantities. Given, the distance between the point from the centre of earth did not change even after the earth shrunk suddenly to half of its present size. Therefore, check if there are any terms that change due to given conditions.
Formula used:
$g=\dfrac{Gm}{{{r}^{2}}}$

Complete answer:
Let us write down the formula of the gravitational constant in terms of known quantities.
$g=\dfrac{Gm}{{{r}^{2}}}$
The value of G is constant always. The mass of the earth is given as constant. It is given in the question that the mass doesn’t change but the size decreases to half.
Also, it is given that the r, the distance of the point from the centre of earth doesn’t change.
Therefore, all the terms in the formula are constant which means the formula is constant.
Hence, the gravitational acceleration doesn’t change and the value will be,
$g=9.8m{{s}^{-2}}$only.

So, the correct answer is “Option A”.

Additional Information:
The gravity of earth, denoted usually as g, is the net acceleration that is imparted to objects due to the combined effect of gravitation and the centrifugal force. The weight of an object on earth's surface is the downwards force on that object, given by Newton's second law of motion. Gravitational acceleration contributes to the total gravity acceleration, but other factors, such as the rotation of Earth, also contribute, and, therefore, affect the weight of the object. Gravity does not normally include the gravitational pull of the Moon and Sun, which are accounted for in terms of tidal effects. It is a vector quantity, and its direction coincides with a plumb bob.

Note:
In the above question, the gravitational acceleration did not change because the mass is kept constant, the distance between the point and the centre of earth is also taken as constant. Therefore, the value did not change as per the given conditions. If the conditions are different, the values change accordingly.