First slide

Gravitational field intensity

Question

If the radius of the earth decreases by 10%, the mass remaining unchanged, what will happen to the acceleration due to gravity?

Difficult

A

Decreases by 19%
B

Increases by 19%
C

Decreases by more than 19%
D

Increases by more than 19%

Solution

$\begin{array}{l} g = \frac{GM}{R^{2}} . . . . . . . . . . (i) \\ g^{'} = \frac{Gm}{{(\frac{90}{100} R)}^{2}} = \frac{100}{81} \frac{GM}{R^{2}} . . . . . . . . . (ii) \end{array}$
From Eqs. (i) and (ii),
$\begin{array}{l} g^{'} = \frac{100}{81} g \Rightarrow \frac{g^{'}}{g} = \frac{100}{81} \\ \frac{g^{'}}{g} - 1 = \frac{100}{81} - 1 \\ ∴ Δg = \frac{19}{81} g = 23 % of g \end{array}$
So increase is more than 19% of g.

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