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1.2.20 · D5Newton's Laws & Dynamics

Question bank — Gravitational field intensity g = GM - r²

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True or false — justify

A heavier test mass sits in a stronger gravitational field.
False. The field depends only on the source and distance ; the test mass cancels in . A feather and a boulder at one point feel identical .
is a scalar because it comes out as a single number like .
False. is a vector; the "" is only its magnitude. Direction matters — near a planet it points inward, and combining fields needs vector addition, not number addition.
At the exact centre of a uniform solid planet, .
True. By symmetry every direction of pull is balanced by an equal pull from the opposite side, so the net field vanishes — even though you are deep inside the mass.
The field intensity and the free-fall acceleration are two different physical quantities that happen to share a number.
False. They are literally equal: and Newton's second law gives , so . Same quantity, two names (field view vs. motion view).
If two planets create fields that point in opposite directions at a point, the net field magnitude is the sum of the two.
False. Opposite directions subtract: . They add only when they point the same way; at the null point they cancel to zero.
becoming zero at a null point means gravity has been "switched off" there.
False. Each source still exerts its full pull; they merely cancel as vectors. A mass placed there feels no net force, but the individual fields are alive.
Units and describe the same thing.
True. , so . The first form reads "force per mass" (field view), the second "acceleration" (motion view).
Doubling the source mass doubles the field, but doubling the distance halves it.
False on the second half. Field is linear in (double → double ) but inverse-square in (double → , not ).

Spot the error

"At altitude , use ."
Error: is measured from the centre of , not the surface. Correct is , so . Using alone blows up wrongly as .
"The field is stronger for the Moon than the Earth because the Moon is smaller, so its lines are more crowded."
Error: Field strength is set by , not by size alone. The Earth's much larger makes its surface field about six times the Moon's despite the larger radius.
"Since has a minus sign, gravity sometimes pushes outward."
Error: The minus is fixed, not "sometimes." points away from ; the minus flips it to point toward — always inward. Gravity is purely attractive here.
"Because , a rocket with no test mass () has an undefined field."
Error: is a property of the location, defined as the limit of for a small test mass; so the ratio stays finite. No test mass need actually be present for the field to exist.
"On the Moon's surface objects fall slower, so is smaller there."
Error: is a universal constant everywhere. The smaller surface comes from the Moon's smaller and in , not from any change in .
"Field lines spreading over explains why ."
Error: The area grows like , so line density (the field) dilutes — it goes like , not . Spreading weakens the field with distance.

Why questions

Why does the test mass disappear from the final formula?
Because the gravitational force is proportional to (), dividing by in cancels it, leaving a quantity that describes only the source and the location.
Why do we bother defining a field instead of just computing force each time?
The field describes the space itself once and for all; then any visiting mass's force is just . It separates "what the source does to space" from "how a visitor responds."
Why is gravity an inverse-square law and not, say, inverse-cube?
Geometrically, a fixed number of field lines pierce every surrounding sphere, and a sphere's area grows as ; the line density (field) therefore falls as . This is the content of Gauss's Law for Gravity.
Why does point toward the source while the unit vector points away?
We choose to point outward by convention, so an attractive pull needs the opposite sign — hence , inward.
Why is the surface value special enough to get its own symbol ?
Because on the surface your distance from the centre is exactly the planet's radius , giving one fixed number we experience daily. See Variation of g with Altitude and Depth for .
Why does the electric field look identical to ?
Both are "source strength over distance squared" fields with the same inverse-square geometry; mass plays the role of charge. Compare in Electric Field Intensity E=kQ/r².

Edge cases

What is as ?
It tends to zero: . Gravity never truly vanishes but fades without limit — it has infinite range but ever-weaker strength.
What happens to as for a point mass?
It diverges to infinity — a mathematical singularity. Real bodies aren't points, so once you go inside, only the mass at smaller radius counts and drops instead (see Variation of g with Altitude and Depth).
Deep inside a uniform planet, does keep rising as you approach the centre?
No. Only the mass in the sphere below you pulls (the outer shell contributes zero net field), so decreases linearly to zero at the centre.
Between two equal masses, where is the null point, and is it stable?
Exactly midway, where the two fields are equal and opposite so . A mass nudged along the line feels a restoring pull back only if the geometry favours it — for two equal point masses the midpoint is unstable along the line.
If a test mass has zero mass, is the concept of meaningless?
No. is defined as the limit of for a vanishingly small test mass; since , the ratio stays well-defined and describes the field even with no visitor present.
Can be non-zero at a point where the gravitational potential is zero?
Yes. Potential and field are independent this way — is the slope of potential, not its value. A flat-but-nonzero potential gives zero field; a zero potential on a slope gives nonzero field. See Gravitational Potential and Potential Energy.

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