CDS (I) 2022 General Knowledge · Question 6 of 120

Correct — C, The force would be halved. Newton's law of universal gravitation says the force between two bodies is F = Gm₁m₂/r², so the two changes described act on it separately and then multiply together. Doubling one mass doubles the numerator and so doubles the force. Doubling the separation divides the force by two squared, that is by four. Combining them, the new force is 2/4 of the original, which is one half. The trap the item is built around is that mass enters to the first power while distance enters squared, so the two doublings do not cancel — they leave a net factor of one half.

• (a)The force would remain the same — The answer if distance entered the formula to the first power, so that a doubling of mass and a doubling of separation cancelled exactly. Distance is squared, and that square is the whole point of an inverse-square law.
• (b)The force would be doubled — Counts the doubled mass and forgets the separation altogether. Moving the bodies apart can only weaken the attraction, so any option that increases the force has ignored half the stem.
• (d)The force would increase by a factor of 4 — Squares the wrong quantity — it applies the factor of four to the mass rather than dividing by it for the distance, and drops the sense of the distance change as well.

Every pair of bodies attracts along the line joining them with a force F = Gm₁m₂/r², where G is the universal gravitational constant, about 6·67 × 10⁻¹¹ N m² kg⁻². The force is proportional to each mass to the first power and inversely proportional to the square of the separation. G is the same everywhere in the universe and must not be confused with g, the acceleration due to gravity, which is local and follows from the law as g = GM/R² for a body of mass M and radius R. The same inverse-square form governs the electrostatic force between charges, and the two are often set side by side for that reason.

Treat a proportionality question one factor at a time and the arithmetic never gets complicated. Write down what each change does on its own — mass doubled gives a factor of 2, separation doubled gives a factor of 1/4 — then multiply, and 2 × 1/4 is 1/2. The direction of each effect is a useful check even before the numbers: adding mass must strengthen the attraction, and adding distance must weaken it, so the answer has to lie between doubling and quartering. Only one option does.

• Newton's law of gravitation is F = Gm₁m₂/r², with G ≈ 6·67 × 10⁻¹¹ N m² kg⁻².
• The force varies as the first power of each mass and as the inverse square of the separation.
• G is a universal constant; g, the acceleration due to gravity, is local and equals GM/R² at the surface of a body.
• Doubling one mass multiplies the force by 2; doubling the separation multiplies it by 1/4; together they give 1/2.
• Coulomb's law for the force between two charges has the same inverse-square form.

• Applying the distance change to the first power and concluding the two effects cancel.
• Confusing the universal constant G with the local acceleration g.
• Assuming a heavier body must fall faster, which the same formula rules out.

As a proportionality item like this one, or as a numerical asking for g on a planet whose mass and radius are given as multiples of the Earth's.

• practice — not a real PYQ

  If the distance between two bodies is reduced to one-third while their masses are unchanged, the gravitational force between them becomes

  • (a)one-third
  • (b)three times
  • (c)nine times
  • (d)one-ninth

  Answer(c) nine times — the force varies as the inverse square of the separation, so dividing the distance by three multiplies the force by three squared.

• practice — not a real PYQ

  A planet has twice the mass of the Earth and twice its radius. The acceleration due to gravity on its surface, compared with that on the Earth, is

  • (a)the same
  • (b)half
  • (c)twice
  • (d)four times

  Answer(b) half — g = GM/R², so doubling M multiplies g by 2 while doubling R divides it by 4, leaving one half.