Gravitational Potential WellJEE Advanced

V(r) and g(r) inside and outside a planet — interactive Physics simulation for IIT-JEE.

Concept

Gravitational potential $V$ is potential energy per unit mass — a well that deepens toward the planet. Outside, $V = -GM/r$ ; inside a uniform planet it keeps falling to $-\tfrac{3}{2}GM/R$ at the centre, while the field $g$ drops linearly to zero there (shell theorem: outer shells pull equally in all directions).

Key formula

V_{out} = -\frac{GM}{r}, \quad V_{in} = -\frac{GM(3R^2 - r^2)}{2R^3}, \quad g_{in} = \frac{GM}{R^3}r, \quad v_{esc} = \sqrt{\frac{2GM}{R}}

Derivation

Outside: integrate $g = GM/r^2$ from ∞. Inside radius r, only the mass $M(r) = M r^3/R^3$ attracts (shell theorem), giving the linear field.

Continue the potential integral inward: $V(r) = V(R) - \int_R^r g\,dr'$ produces the parabolic cap. Note $V$ and $g$ are both continuous at the surface; $g$ peaks exactly there.

Scenarios to explore

Gravitational Potential Well — V(r) and g(r) inside & outside a planet — shell theorem.

Real-world applications

Escape velocity and gravitational binding of planets.
Tunnel-through-Earth SHM thought experiment (T ≈ 84 min).
Structure of stars — pressure balances the potential gradient.

JEE exam tips

V_centre = 1.5 × V_surface (both negative) for a uniform sphere — instant recall.
A ball dropped through a diametric tunnel executes SHM with ω² = GM/R³ = g/R.
Escape speed is √2 × orbital speed at the surface.

Common mistakes

Using −GM/r inside the planet.
Thinking g is maximum at the centre — it is ZERO there; V is at its minimum.
Sign slips: V is negative everywhere, and MORE negative closer in.

Exam traps to avoid

g inside depends only on the mass BELOW you; V depends on ALL the mass.
Escape velocity is independent of launch direction (energy argument).