Question

If the radius of the earth shrinks by 2.0%, mass remaining constant, then how would the value of acceleration due to gravity changes?
1. 4%
2. 2%
3. 8%
4. None

Answer 1

Correct Answer - Option 1 : 4%

Let the mass of the body is m and mass of earth is M.

Gravitational force between the bodies, \(F = \frac{{GMm}}{{{R^2}\;}}\)

Acceleration due to gravity, \(g = \frac{F}{m} = \frac{{GM}}{{{R^2}}}\)

Taking logarithm on both sides, we get

\(\log g = \log G + \log M - 2\log R\)

As G and M are constant, so differentiation of the above equation gives

\(\frac{{dg}}{g} = 0 + 0 - 2\frac{{dR}}{R}\)

Now as the radius of earth decreases by 2%,

\(\frac{{dR}}{R} = \; - \frac{2}{{100}}\)

\(\frac{{dg}}{g} \times 100 = - 2 \times \frac{{dR}}{R} \times 100 = - 2 \times \left( { - \frac{2}{{100}}} \right) \times 100 = 4\% \)