Gravitational FieldJEE Main

g inside and outside a planet — interactive Physics simulation for IIT-JEE.

Concept

Gravity is strongest at a planet's surface. Go outward and it falls off as $1/r^2$ ; go inward and it falls linearly to zero at the centre — because only the mass beneath you pulls (the shell theorem).

Key formula

g_{out} = \frac{GM}{r^2}\;(r\ge R), \qquad g_{in} = \frac{GMr}{R^3}\;(r<R)

Derivation

Outside a uniform sphere, all its mass acts as if concentrated at the centre, giving $g = GM/r^2$ .

Inside, only the sphere of radius $r$ beneath you contributes (the outer shell's pull cancels). Its mass scales as $r^3$ , so $g \propto r^3/r^2 = r$ — straight-line growth from zero at the centre to $g_{surface}$ at $r=R$ .

Scenarios to explore

Gravitational Field — g inside and outside a planet (shell theorem).

Real-world applications

Variation of weight with altitude and depth.
Modelling planetary interiors and tides.
Satellite and probe trajectory design.

JEE exam tips

At depth $d$ : $g = g_s(1 - d/R)$ . At height $h \ll R$ : $g \approx g_s(1 - 2h/R)$ .
g is continuous at the surface — both formulas give $GM/R^2$ there.

Common mistakes

Using $1/r^2$ for points inside the planet — there it is linear in $r$ .
Forgetting g peaks exactly at the surface, not the centre.
Mixing up field $g$ with potential $V$ (which is most negative at the centre).

Exam traps to avoid

Weight is zero at a planet's centre, not maximum.
Doubling altitude does not halve g — it follows the inverse-square law from the centre.