Question

If the radius of the earth is reduced to half of its present value, with no change in the mass, the acceleration due to gravity will be:
A. Doubled to that of the original value
B. Four times to that of the original value
C. Remains same
D. Reduced to half of its original value

Answer 1

Hint: In these types of questions, we always first write the formula for the original value of the quantity in consideration, here it is acceleration due to gravity, then we make the changes given in the question and then compare it with the original formula. Here, we need to keep a formula in mind which is:
\[g=\dfrac{GM}{{{R}^{2}}}\]

Complete step-by-step answer:
The formula for the acceleration due to gravity (g) is given as \[g=\dfrac{GM}{{{R}^{2}}}\]​ ...(1)
Where G is universal gravitational constant and M is mass of the earth with R as the radius of the earth.
Now replacing R by \[\dfrac{R}{2}\]in the equation-1
we get the new value of ‘g’ as

\[{{g}_{new}}=\dfrac{GM}{\dfrac{{{R}^{2}}}{4}}=4\dfrac{GM}{{{R}^{2}}}\]​
Which turns out to be four times of its original value.
Hence, the correct answer is that If the radius of the earth is reduced to half of its present value, with no change in the mass, the acceleration due to gravity will become Four times to that of the original value which is option B.

Note: Gravity or gravitational force can be defined as a natural phenomenon by which all things with mass or energy including planets, stars, galaxies, and even light are brought towards one another by forces of attraction. On Earth, this gravity gives weight to physical objects and matter, and the Moon's gravity causes the ocean tides.