MHT-CET Physics · Gravitation

Newton's Law and the Field of a Sphere

Every two masses attract with a force Gm₁m₂/r² along the line joining them; outside a uniform sphere or shell the field is as if all the mass sat at the centre, inside a shell it is zero, and inside a solid sphere it grows in proportion to the distance from the centre.

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Why this matters

7 PYQs, 3 of them HARD: the field inside and outside a uniform sphere, a sphere inside a shell, three masses at the corners of a triangle, a ring's pull on a mass at its centre, and how the force between two touching spheres scales with size. One card.

Concept 1 of 1: Inverse-Square Force, Shells and Spheres

The force Gm₁m₂/r² acts along the line joining the centres, like the electrostatic force, but it only ever attracts. For symmetric shapes, add the pulls as vectors: a ring or a symmetric arrangement cancels at its centre, so a mass at the centre of a ring feels nothing. Outside any spherically symmetric body the field is GM/r², all the mass at the centre; inside a thin shell it is zero; inside a uniform solid sphere only the mass nearer the centre counts, and the field is GMr/R³, growing linearly. For two touching spheres of radius R and density ρ, the mass goes as R³ρ and the separation as R, so the force goes as R⁴ρ².

Definition

$F = \dfrac{Gm_1m_2}{r^2}$ , along the line joining the masses, always attractive.
Outside a sphere or shell: $g = \dfrac{GM}{r^2}$ . Inside a shell: 0. Inside a solid sphere: $g = \dfrac{GMr}{R^3}$ .
Sphere of mass m in a shell of mass m and radius 2r: at 2.5r, $g = \dfrac{G(2m)}{(2.5r)^2} = \dfrac{8}{25}\dfrac{Gm}{r^2}$ .
Ring's pull at its centre: 0. Triangle of side L/3, mass at a midpoint: only the far corner pulls, $\dfrac{12Gm_1m_2}{L^2}$ .
Touching spheres: $F \propto R^4\rho^2$ .

Newton's law and the sphere

F = \frac{Gm_1m_2}{r^2}, \qquad g_{\text{out}} = \frac{GM}{r^2}, \qquad g_{\text{in}} = \frac{GMr}{R^3}

Worked example

A uniform sphere of radius R has field g₀ at its surface. What is the field at R/2 inside it and at 2R outside it?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 2nd May Shift 1 · Q41Hard

Example 1 · Gravitation · Newton's Law of Gravitation and Gravitational Force

The magnitude of gravitational field at distance

r_1

and

r_2

from the centre of a uniform sphere of radius

R

and mass

M

are

F_1

and

F_2

respectively. The ratio

F_1/F_2

will be (if

r_1 > R

and

r_2 < R

)

Using GM/r² inside a solid sphere

Inside, only the mass closer to the centre pulls, and the field falls to zero at the centre: g = GMr/R³. The ratio of an outside and an inside field is R³/(r₁²r₂).

Forgetting the shell's mass outside it

Inside a shell its own pull cancels, but outside it the shell's mass adds to everything within. Between a sphere and a surrounding shell only the sphere counts; beyond the shell both do.

Drill 6 more on inverse-square force, shells and spheres

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Inverse-Square Force, Shells and Spheres
Newton's law and the sphere
$F = \frac{Gm_1m_2}{r^2}, \qquad g_{\text{out}} = \frac{GM}{r^2}, \qquad g_{\text{in}} = \frac{GMr}{R^3}$

Watch out for (2)

Using GM/r² inside a solid sphere
→ Inverse-Square Force, Shells and Spheres
Forgetting the shell's mass outside it
→ Inverse-Square Force, Shells and Spheres

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