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The dependence of acceleration due to gravity g on the distance r from the centre of the earth, assumed to be a sphere of radius R of uniform density, is as shown in figures below.
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The correct figure is
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The correct figure is
A
(d)
B
(a)
C
(b)
D
(c)
Detailed Solution
Inside the earth (r < R), only the mass within radius r attracts: $g = \frac{GM_r}{r^2}$ with $M_r = M\frac{r^3}{R^3}$, so $g = \frac{GM}{R^3}r$
Thus inside the earth $g \propto r$: it is zero at the centre and increases linearly up to the surface.
At the surface (r = R) it has its maximum value $g = \frac{GM}{R^2}$.
Outside the earth (r > R): $g = \frac{GM}{r^2}$, so $g \propto \frac{1}{r^2}$; it falls off as a curve and tends to zero at large r.
The correct graph is therefore a straight line from the origin up to r = R, followed by a curve decreasing as $\frac{1}{r^2}$; this is the figure labelled (d).
Thus inside the earth $g \propto r$: it is zero at the centre and increases linearly up to the surface.
At the surface (r = R) it has its maximum value $g = \frac{GM}{R^2}$.
Outside the earth (r > R): $g = \frac{GM}{r^2}$, so $g \propto \frac{1}{r^2}$; it falls off as a curve and tends to zero at large r.
The correct graph is therefore a straight line from the origin up to r = R, followed by a curve decreasing as $\frac{1}{r^2}$; this is the figure labelled (d).