Question

If the radius of the earth is reduced to half of its present value, with no change in the mass,how will the acceleration due to gravity ,be affected?

Answer 1

acceleration will double

Answer 2

Answer:

The acceleration due to gravity will be affected by a factor of 4 which increases.

Explanation:

Given is the radius of earth be ""reduced to half"" of its present value, the mass change is neglected then the ""acceleration due to gravity"" also changes by the following measure -

Given is the $R^{\prime}=\frac{R}{2}$

As we know that F=$\bold{\frac{G M m}{R^{2}}}$

Since F=m $\bold{\times}$ a

$\begin{array}{l}{m \times a=\frac{G M m}{R^{2}}} \\ {a=\frac{G M}{R^{2}}}\end{array}$

Here, G is the Newtonian gravitational constant

M is the Mass of the earth

R is the ""radius of the Earth""

a is the acceleration due to gravity

Thereby it is proportional to $\frac{1}{R^{2}}$  

$\bold{R^{\prime}=\frac{R}{2}}$

$a^{\prime} proportional to \frac{1}{\left(R^{\prime}\right)^{2}}$

$a^{\prime} proportional to \frac{1}{\left(\frac{R}{2}\right)^{2}}$  

$a^{\prime} proportional to \frac{\frac{1}{R^{2}}}{4}$

$a^{\prime} proportional to \frac{4}{R^{2}}$

$a^{\prime}= G M\left(\frac{4}{R^{2}}\right)$

$a^{\prime}a' = 4 a$

Therefore, the ""acceleration due to gravity"" increases by a ""factor of 4"".