GRE Physics Companion: Newton's Shell Theorem and Spherical Mass Distributions

GRE Physics Companion: Newton’s Shell Theorem and Spherical Mass Distributions

For a thin spherical shell,

(1)

For a general spherical distribution,

(2)

The vector points inward.

1 High-value GRE facts

1. A uniform thin spherical shell produces zero field everywhere inside.
2. Outside a spherical shell, the field is exactly the point-mass field.
3. For spherical matter, only enclosed mass contributes to the interior field.
4. M_enc(r) = 4π ∫ ₀^rρ(s)s²ds.
5. Uniform-sphere enclosed mass scales as r³.
6. Uniform-sphere interior field scales as r.
7. The field of a uniform solid sphere is continuous at the surface.
8. A concentric hollow region has zero field.
9. A regular spherical density gives g ∝ r near the center.
10. Spherical symmetry, not uniform density, is what guarantees exact exterior point-mass behavior.

Part I: Original GRE-style problems

Problem 1: field inside a shell

At a point inside a uniform thin spherical shell, the net gravitational field is

1. GM∕r²
2. GM∕R²
3. zero
4. infinite
5. proportional to r

Problem 2: field outside a shell

At r > R, a uniform thin spherical shell of total mass M produces

1. GM∕R²
2. GM∕r²
3. zero
4. GMr∕R³
5. 2GM∕r²

Problem 3: uniform sphere enclosed mass

For a uniform solid sphere of radius R and total mass M, the mass enclosed within r < R is

1. M(r∕R)
2. M(r∕R)²
3. M(r∕R)³
4. M(R∕r)²
5. M

Problem 4: uniform sphere interior field

Inside a uniform solid sphere,

(3)

is proportional to

1. 1∕r²
2. 1∕r
3. a constant
4. r
5. r²

Problem 5: field at the center

At the exact center of any regular spherically symmetric mass distribution, The Gravitational Field is

1. infinite
2. zero
3. GM∕R²
4. determined only by surface density
5. undefined

Problem 6: half-radius uniform sphere

For a uniform sphere, the field magnitude at r = R∕2 is what fraction of the surface field?

1. 1∕8
2. 1∕4
3. 1∕2
4. 1
5. 2

Problem 7: hollow concentric cavity

Inside the empty concentric cavity of a uniform thick spherical shell, the field is

1. zero
2. proportional to 1∕r²
3. proportional to r
4. constant and nonzero
5. infinite

Problem 8: volume shell mass

A thin spherical volume shell of radius r and thickness dr has mass

1. ρdr
2. 2πρr dr
3. 4πρr² dr
4. (4∕3)πρr³ dr
5. ρr² dr

Problem 9: exterior nonuniform sphere

A spherically symmetric body has nonuniform density but total mass M. At r > R, its field magnitude is

1. zero
2. GM∕r²
3. dependent on the detailed radial density profile
4. GM∕R²
5. proportional to r

Problem 10: shell surface jump

For an ideal thin shell with surface density Σ, the magnitude of the one-sided field jump is

1. GΣ
2. 2πGΣ
3. 4πGΣ
4. GM∕R
5. zero

Problem 11: general enclosed mass

For spherical density ρ(r),

(4)

is

1. 4π ∫ ₀^rρ(s)s²ds
2. 2π ∫ ₀^rρ(s)sds
3. ∫ ₀^rρ(s) ds
4. 4πr²ρ(r)
5. ρ(r)(4πr³)

Problem 12: small-radius scaling

If the central density is finite and nonzero, the gravitational field sufficiently near the center scales as

1. 1∕r²
2. 1∕r
3. constant
4. r
5. r³

Part II: Complete worked solutions

Solution 1

Newton’s shell theorem gives exactly zero field everywhere inside a uniform thin spherical shell. Answer: (C).

Solution 2

Outside the shell, the shell acts exactly like a point mass at its center:

(5)

Answer: (B).

Solution 3

For constant density, enclosed mass follows enclosed volume:

(6)

Answer: (C).

Solution 4

Substituting M_enc ∝ r³ into g = GM _enc∕r² gives g ∝ r. Answer: (D).

Solution 5

Symmetry requires the field to vanish at the center, and a finite central density gives a regular field there. Answer: (B).

Solution 6

For a uniform sphere,

(7)

At r = R∕2, this is 1∕2. Answer: (C).

Solution 7

All mass shells lie outside points in the concentric cavity, so each shell contributes zero field. Answer: (A).

Solution 8

The thin-shell volume is

(8)

so

(9)

Answer: (C).

Solution 9

Spherical symmetry guarantees the exact exterior point-mass field regardless of radial density variation. Answer: (B).

Solution 10

For a shell,

(10)

and

(11)

Answer: (C).

Solution 11

The spherical volume element is

(12)

after angular integration. Answer: (A).

Solution 12

For finite ρ(0),

(13)

so

(14)

Answer: (D).

2 GRE checklist

1. Decide first whether the field point is inside or outside each shell.
2. Use enclosed mass, not total mass, for interior spherical fields.
3. For uniform density, remember M_enc ∝ r³ and g ∝ r.
4. For exterior points, spherical symmetry makes the exact field GM∕r².
5. Check continuity for volume densities and the jump for ideal surface shells.
6. Use 4πr²dr for spherical shell volume.

References

References