Question

If the radius of the earth suddenly decreases to $80\%$ of its present value, the mass of the earth remaining the same, the value of the acceleration due to gravity will.

• A. Remain unchanged
• B. Become $9.8/0.64$ m $s^{-2}$
• C. Increase by $36\%$
• D. Increase by about $56\%$

Answer 1

Answer: (D)

Increase by about $56\%$

The expression for acceleration due to gravity can be derived from Newton's law of gravitation and it says that the acceleration due to gravity is

$g=\frac{GM}{R^2}$

where G is the universal gravitational constant, M is the mass of earth, R is the radius of the earth.

If R decreases to 80% of R, the new radius is R'=0.8R. Since G and M remain constants, we have the new acceleration due to gravity as

$g^{\prime }=\frac{GM}{R^{\prime 2}}$

put the value of R' = 0.8R

$g^{\prime }=1.56\frac{GM}{R^2}$

g' = 1.56g

hence the value of g increases by 56%