As an object moves away from the surface of the Earth, the graph of the magnitude of the gravitational force versus r, the distance from the surface R, is:Single choice
A
a.
B
b.
C
c.
D
d.
E
e.

Question

As an object moves away from the surface of the Earth, the graph of the magnitude of the gravitational force versus r, the distance from the surface R, is:

BlackTom AI · Accepted Answer

Newton’s universal law of gravitation: F = G M m / d^2. Here d is the distance from the center of Earth; if r is measured from the surface then d = R + r, so the force as a function of r is
F(r) = G M m / (R + r)^2.

1) Relevant concepts
- Gravitational force follows an inverse-square law (∝ 1/d^2).
- If r is distance from the surface, the denominator is (R + r)^2.
- Behavior: at r = 0 the force equals GMm/R^2 (a finite positive value), and as r → ∞ the force → 0.

2) Step-by-step reasoning and intermediate calculus
- Evaluate at the surface: F(0) = GMm/R^2 (nonzero).
- Monotonicity: dF/dr = d/dr [GMm (R + r)^{-2}] = -2 GMm (R + r)^{-3} < 0 for r ≥ 0. So F decreases for increasing r.
- Concavity: d^2F/dr^2 = 6 GMm (R + r)^{-4} > 0, so the curve is concave upward (slope becomes less negative as r increases).
- Limit at infinity: lim_{r→∞} F(r) = 0. Thus the graph starts at a finite height at r = 0, decreases monotonically, flattens out and approaches zero asymptotically.

3) Why option (a) is correct
- Option (a) shows a curve that begins at a finite positive value at r = 0, falls off rapidly at small r, and levels off approaching the horizontal axis as r increases — exactly the qualitative shape of F(r) = GMm/(R + r)^2 with the concavity noted above.

4) Why the other choices are incorrect
- (b) shows a straight linear decrease to zero — wrong because gravitational force is not linear in r.
- (c) and (e) show an increasing force with r — opposite of the inverse-square behavior.
- (d) shows an increasing curve that flattens — again incorrect direction.

Conclusion: The inverse-square dependence F(r) = GMm/(R + r)^2 produces the decreasing asymptotic curve in (a). Therefore the correct answer is a.

As an object moves away from the surface of the Earth, the graph of the magnitude of the gravitational force versus r, the distance from the surface R, is:Single choice

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