GRE Physics Companion: Gravitational Potential Energy and Gravitational Potential

GRE Physics Companion: Gravitational Potential Energy and Gravitational Potential

For point-mass gravity,

(1)

The field is related to potential by

(2)

1 High-value GRE facts

1. With zero at infinity, U = −GMm∕r.
2. gravitational potential is Φ = U∕m = −GM∕r.
3. g = −∇Φ.
4. Potential is scalar and superposes algebraically.
5. A shell has zero interior field but nonzero constant interior potential.
6. A uniform sphere has a quadratic interior potential.
7. Near a surface, ΔU ≈ mgh only for small height changes.
8. Escape corresponds to the threshold total energy E = 0.
9. v_esc = .
10. For a system of masses, count each interacting pair once.

Part I: Original GRE-style problems

Problem 1: sign of point-mass potential energy

With U(∞) = 0, the gravitational potential energy of two positive masses at finite separation is

1. positive
2. zero
3. negative
4. imaginary
5. undefined

Problem 2: gravitational potential

The gravitational potential of a point mass M is

1. −GM∕r
2. −GMm∕r
3. GM∕r²
4. −GM∕r²
5. GMm∕r

Problem 3: field from potential

The Gravitational Field is related to gravitational potential by

1. g = ∇Φ
2. g = −∇Φ
3. g = Φr
4. g = ∇²Φ
5. g = 0 everywhere

Problem 4: interior shell potential

Inside a thin uniform spherical shell of radius R and mass M, the gravitational potential is

1. zero
2. −GM∕r
3. −GM∕R
4. −GMr∕R²
5. +GM∕R

Problem 5: uniform-sphere center potential

At the center of a uniform solid sphere,

(3)

equals

1. −GM∕R
2. −3GM∕(2R)
3. −GM∕(2R)
4. zero
5. −2GM∕R

Problem 6: near-surface approximation

The approximation

(4)

is best when

1. h ≫ R
2. h is comparable to R
3. |h|≪ R
4. g = 0
5. the source is nonspherical

Problem 7: escape speed

The minimum Newtonian escape speed from radius r is

1. 
2. 
3. GM∕r
4. 2GM∕r
5. 

Problem 8: circular versus escape speed

At the same radius,

(5)

is

1. 1∕2
2. 1
3. 
4. 2
5. 2π

Problem 9: equipotential geometry

The equipotential surfaces of an isolated point mass are

1. planes
2. cylinders
3. spheres
4. cones
5. helices

Problem 10: zero field

At a point where the gravitational field is zero, the gravitational potential

1. must also be zero
2. must be positive
3. need not be zero
4. must be infinite
5. cannot be defined

Problem 11: scalar superposition

For several point masses, gravitational potential is found by

1. vector addition of each potential
2. scalar addition of each potential
3. multiplying the individual potentials
4. averaging the masses first
5. adding only the nearest mass

Problem 12: pair counting

For three point masses, the number of distinct pairwise gravitational-potential-energy terms is

1. 1
2. 2
3. 3
4. 6
5. 9

Part II: Complete worked solutions

Solution 1

With zero at infinity,

(6)

for finite positive r. Answer: (C).

Solution 2

Potential is potential energy per unit test mass:

(7)

Answer: (A).

Solution 3

A conservative gravitational field points toward decreasing potential:

(8)

Answer: (B).

Solution 4

The shell interior field is zero, so the potential is constant and equals its surface value:

(9)

Answer: (C).

Solution 5

For a uniform sphere,

(10)

Answer: (B).

Solution 6

The constant-g approximation is accurate when the height change is small compared with the body’s radius. Answer: (C).

Solution 7

Set the minimum escape energy equal to zero:

(11)

Thus

(12)

Answer: (B).

Solution 8

Since

(13)

(14)

Answer: (C).

Solution 9

Point-mass potential depends only on radius, so constant-potential surfaces are spheres. Answer: (C).

Solution 10

Zero field means zero gradient of potential, not necessarily zero potential itself. Answer: (C).

Solution 11

Potential is a scalar, so individual potentials add algebraically. Answer: (B).

Solution 12

The distinct pairs are (1, 2), (1, 3), and (2, 3). Answer: (C).

2 GRE checklist

1. Keep U and Φ distinct.
2. Use the minus sign in both U = −GMm∕r and g = −∇Φ.
3. Remember that zero field does not imply zero potential.
4. Use mgh only as a local approximation.
5. Set E = 0 for the minimum escape condition.
6. Count each pair once in system potential energy.

References

References