A non-homogenous sphere of radius a has the following density (d) variations.
$\begin{array}{l} d = d_{0} w h e r e r \leq \frac{a}{3} \\ d = \frac{d_{0}}{2} w h e r e \frac{a}{3} < r \leq a \end{array}$ (r is the distance from the centre of the sphere)
Consider the following statements:
1) Gravitational field at r = $\frac{a}{2}$ is greater than that r=2a.
2) Gravitational field at r = $\frac{a}{4}$ is greater than that at r=2a
Which of the statements given above is/are correct?

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Neither 1 nor 2

2 only

1 only

Both 1 and 2

answer is C.

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Detailed Solution

$\begin{array}{l} Gravitational field at (r = \frac{a}{2}) = \frac{4}{3} π {(\frac{a}{3})}^{3} d_{o} + (\frac{4}{3} π {(\frac{a}{2})}^{3} - \frac{4}{3} π {(\frac{a}{3})}^{3}) \times \frac{d_{o}}{2} \\ = \frac{35 {Gd}_{0} a}{81} = 0.43 K \\ Similarly \\ Gravitational field at (r = 2 a) = \frac{14 {Gd}_{0} a}{81} = 0.17 K \\ Gravitational field at (r = \frac{a}{4}) = \frac{{Gd}_{0} πa}{3} = 0.33 K \end{array}$