CDS (II) 2020 General Knowledge · Question 87 of 120

Correct — D, decrease by four times. Newton's law of gravitation gives F = Gm₁m₂/r², so the force falls as the square of the separation. Increase r by a factor of two and the denominator grows by 2² = 4, so F becomes one-quarter of what it was. The paper's phrase 'decrease by four times' is loose English for 'become one-fourth', and it is the intended reading; the same key would be expressed in a physics text as F' = F/4. Note that the masses do not enter the change at all, since neither of them alters — only the distance does, and the inverse-square dependence does the rest.

• (a)remain same — The force cannot stay the same, because r appears in the formula and r has changed. It would remain the same only if a doubling of distance were offset by a fourfold increase in one of the masses.
• (b)increase by two times — An increase of any kind is impossible when the bodies move further apart. Gravitational attraction weakens with distance, never strengthens.
• (c)decrease by two times — This is the answer you get from an inverse-first-power law. Gravitation is inverse-square, so doubling the distance divides the force by four, not by two.

Newton's law of universal gravitation states that every particle attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres, F = Gm₁m₂/r². The constant G is about 6.674 × 10⁻¹¹ N m²/kg². The force is always attractive, acts along the line joining the two bodies, obeys Newton's third law so that each pulls on the other equally, and is independent of the medium between them. It is by far the weakest of the four fundamental forces, but it dominates on astronomical scales because mass comes in only one sign and so gravity never cancels.

Every item of this family is answered by tracking the exponents. Mass appears to the first power in the numerator, so doubling one mass doubles the force; distance appears squared in the denominator, so doubling the distance quarters it. Do both together and the two effects multiply — force doubled and then quartered leaves it halved, which is exactly what a companion CDS question asks. The inverse-square form is not peculiar to gravity: Coulomb's law for electric charges has the same shape, and both follow from the geometry of influence spreading over the surface of a sphere.

• F = Gm₁m₂/r², with G about 6.674 × 10⁻¹¹ N m²/kg².
• Doubling the separation reduces the force to one-quarter; halving it multiplies the force by four.
• Doubling one mass while doubling the distance leaves the force halved.
• Gravitation is always attractive, acts along the line joining the centres, obeys Newton's third law and is independent of the intervening medium.
• Coulomb's electrostatic law has the same inverse-square form, but electric force can be attractive or repulsive.

• Treating the law as inverse-first-power and halving the force instead of quartering it.
• Reading 'decrease by four times' as anything other than becoming one-fourth.
• Assuming an intervening medium or shield can reduce gravitational attraction. It cannot.

As a proportionality item varying mass or distance, or as a numerical using F = Gm₁m₂/r².

• practice — not a real PYQ

  If the distance between two bodies is halved, the gravitational force between them becomes

  • (a)half
  • (b)double
  • (c)four times
  • (d)one-fourth

  Answer(c) four times — halving r divides the denominator by four, so the force is multiplied by four.

• practice — not a real PYQ

  The gravitational force between two bodies is

  • (a)reduced when a thick metal sheet is placed between them
  • (b)independent of the medium between them
  • (c)repulsive at very large distances
  • (d)greater on the heavier body than on the lighter one

  Answer(b) independent of the medium between them — gravitation cannot be screened, is always attractive, and the two bodies pull on each other equally.