NDA (I) 2017 General Ability Test · Question 123 of 149

Correct — D, The quantity G is a universal constant. Newton's law of gravitation says that any two point masses attract each other with a force proportional to the product of the masses and inversely proportional to the square of the distance between them, and G is the constant of proportionality that turns that statement into an equation. Its value, about 6.674 × 10⁻¹¹ newton metre squared per kilogram squared, is the same for every pair of bodies, at every separation, in every part of the universe — which is exactly what calling it universal means. Cavendish first measured it in 1798 with a torsion balance, and no experiment since has found it to vary with place, time or the nature of the masses involved.

• (a)The quantity G depends on the local value of g, acceleration due to gravity — The dependence runs the other way. For a body of mass M and radius R, the surface value of g works out as GM/R², so g is calculated from G and not the reverse. That is why g differs from planet to planet and even from place to place on Earth, while G stays the same.
• (b)The quantity G is greatest at the surface of the Earth — This confuses G with g. The acceleration due to gravity is indeed largest at the surface and falls off both above it and below it, but G does not vary with position at all — there is no location at which it is greater or smaller.
• (c)The quantity G is used only when earth is one of the two masses — The law is universal in its reach as well as in its constant. It applies to the Sun and a planet, to two stars, to the Moon and a tide, and to two ordinary objects on a laboratory bench — which is precisely the arrangement Cavendish used to measure G in the first place, with no planet involved.

Two quantities in gravitation are constantly confused. G is the universal gravitational constant, the same everywhere, with dimensions M⁻¹L³T⁻² and units of newton metre squared per kilogram squared. Small g is the acceleration due to gravity at a particular place, measured in metres per second squared, and it is a local quantity — it depends on the mass and radius of the body you are standing on, and on latitude, altitude and depth. The bridge between them is g = GM/R², which shows at once why one is fixed and the other is not.

Three of the four options here are variations on the same misunderstanding, that G behaves like g. The way to keep them apart is to remember what each one is for. G converts masses and distance into a force and belongs to the law itself; g describes the effect of one particular body at one particular spot. Since Earth's shape is not a perfect sphere and it spins, g is slightly larger at the poles than at the equator — a variation UPSC has tested directly — and yet through all of that G does not move at all. If a statement makes G change with place, position or the identity of the masses, it is wrong.

• The universal gravitational constant G is about 6.674 × 10⁻¹¹ N m² kg⁻² and has the dimensions M⁻¹L³T⁻².
• Acceleration due to gravity at the surface of a body is g = GM/R², so g varies while G does not.
• Henry Cavendish measured G in 1798 using a torsion balance with laboratory masses, not planetary ones.
• On Earth g is slightly larger at the poles than at the equator, because of the equatorial bulge and the planet's rotation.

• Treating G as though it were g and letting it change with place.
• Assuming the law applies only when one of the bodies is a planet.
• Confusing mass, which does not change with location, with weight, which does.

NDA asks for the unit, the dimension or the constancy of G, or for the way g varies over the Earth.

• practice — not a real PYQ

  The value of the universal gravitational constant G

  • (a)is largest at the surface of the Earth
  • (b)is the same everywhere in the universe
  • (c)changes from planet to planet
  • (d)becomes zero at the centre of the Earth

  Answer(b) is the same everywhere in the universe — it is g, not G, that varies from place to place and body to body.

• practice — not a real PYQ

  A body is taken from the Earth to the Moon. Which one of the following remains unchanged?

  • (a)Its weight
  • (b)The acceleration due to gravity acting on it
  • (c)Its mass
  • (d)The force of gravity on it

  Answer(c) Its mass — mass is an intrinsic property, while weight and the local value of g both change.