NDA (II) 2016 General Ability Test · Question 78 of 150

Correct — C, the Earth is approximately an ellipsoid whose equatorial radius is the larger one, by about 21 km. Acceleration due to gravity at the surface goes as GM/R², so it depends inversely on the square of the distance from the centre of the Earth. The equatorial radius is roughly 6,378 km and the polar radius roughly 6,357 km, a difference of about 21 km, which means a person standing at the equator is that much further from the centre than a person at a pole. Being further out, the equatorial observer sits where the same mass pulls a little more weakly, so g there is smaller — about 9.78 m/s² at the equator against about 9.83 m/s² at the poles. Walking from the equator towards either pole therefore takes you closer to the centre and g rises, which is precisely what the stem describes.

• (a)Earth is a sphere with same density everywhere — This describes a perfectly uniform sphere, and on such a body the surface value of g would be identical at every point. It is the one model that positively forbids the variation the question asks you to explain.
• (b)Earth is a sphere with different density at the polar regions than in the equatorial regions — The Earth's interior is layered by depth, not split into a polar half and an equatorial half, and no such systematic side-to-side density contrast is what drives the equator-to-pole change in g. Local rock density does produce small gravity anomalies that surveyors measure, but they are far too small and too patchy to give the smooth latitude trend in the stem.
• (d)Earth is approximately an ellipsoid having its equatorial radius smaller than its polar radius by 21 km — This gets the flattening exactly backwards. A body spinning on its axis bulges outwards at the equator and flattens at the poles, so the equatorial radius must be the bigger one. With the radii swapped as this option has them, g would come out larger at the equator, the opposite of the observed fact the stem starts from — which is why it is the trap for anyone who reads only the number 21 and not the direction.

The Earth is an oblate spheroid: it rotates, and the rotation throws material outwards most strongly at the equator, giving an equatorial bulge and a polar flattening. The equatorial radius is about 6,378 km, the polar radius about 6,357 km, so the equator sits about 21 km further from the centre. Since surface gravity varies as one over the square of the distance from the centre, g is smallest at the equator and largest at the poles.

Two separate effects push g down at the equator, and it is worth keeping them apart. The first is the one this option names — the equatorial surface is simply further from the centre. The second is that rotation supplies a centripetal requirement, so part of the true gravitational pull is used up keeping an equatorial object turning with the Earth, and the effective free-fall acceleration is reduced further; that effect is zero at the poles, where an object sits on the rotation axis. The rotation is also what created the bulge in the first place, so the two are related rather than independent. The examiner has offered only the shape as an option, and the shape is genuinely the dominant part of the roughly 0.05 m/s² equator-to-pole difference, so option (c) is the right pick. The discrimination that actually decides this item is between (c) and (d), and it is only about which radius is larger.

• Earth's equatorial radius is about 6,378 km and its polar radius about 6,357 km, a difference of roughly 21 km.
• g is about 9.78 m/s² at the equator and about 9.83 m/s² at the poles; the standard value 9.8 m/s² sits between them.
• Surface gravity varies as GM/R², so a larger radius at the same mass means a smaller g.
• Rotation also lowers the effective value of g at the equator and has no such effect at the poles, because a pole lies on the axis of rotation.

• Swapping the two radii — the equatorial radius is the larger one, because rotation makes the Earth bulge at the equator.
• Crediting the whole equator-to-pole change in g to rotation alone, or to the shape alone; both contribute, with the shape dominant.

This appears either as a straight reason-for-variation item like this one or as an assertion-and-reason pair about weight changing with latitude, so learn the direction of the flattening, not just the 21 km figure.

• practice — not a real PYQ

  The weight of a body is maximum at which one of the following locations on the Earth's surface?

  • (a)At the equator
  • (b)At either pole
  • (c)At 45 degrees latitude
  • (d)It is the same everywhere

  Answer(b) At either pole — the poles are closer to the Earth's centre and lie on the rotation axis, so g and therefore weight are greatest there.

• practice — not a real PYQ

  The shape of the Earth is best described as

  • (a)a perfect sphere
  • (b)a prolate spheroid elongated along the equator
  • (c)an oblate spheroid flattened at the poles
  • (d)a cylinder with hemispherical caps

  Answer(c) an oblate spheroid flattened at the poles — rotation produces an equatorial bulge and polar flattening.