Which of the following statements is/are correct? 1. Angular velocity for all locations on the Earth’s surface is the same while linear velocity varies. 2. Linear velocity is maximum at the equator and minimum at the poles. Select the correct answer using the code given below.
- (a)1 only
- (b)2 only
- (c)Both 1 and 2
- (d)Neither 1 nor 2
Correct — C, both 1 and 2. The Earth turns as a single rigid body, so every point on its surface sweeps out the same angle in the same time — one full turn a day, an angular velocity of about 7.29 by ten to the power minus five radians per second, identical at Kanyakumari and at Leh. Linear velocity is a different matter, because it is the actual distance covered per second, and that depends on how far you stand from the axis of rotation. A point at the equator traces the largest circle and moves fastest, at roughly 1,675 kilometres an hour; at 45 degrees latitude the circle is smaller and the speed drops to about 1,184 kilometres an hour; at the pole the point simply spins in place and the linear speed is effectively zero. The rule is that linear speed equals the equatorial speed multiplied by the cosine of the latitude. That makes statement 1 correct and statement 2 correct.
- (a)1 only — Rejects statement 2, but linear velocity really is greatest at the equator and falls to zero at the poles, because the radius of the circle traced shrinks with latitude.
- (b)2 only — Rejects statement 1, but angular velocity is the same everywhere on the surface — the whole Earth completes one rotation in the same time.
- (d)Neither 1 nor 2 — Both statements are standard consequences of rigid-body rotation; there is nothing wrong with either.
For a rotating rigid body, angular velocity is the rate of turning and is a property of the whole body, while linear or tangential velocity is the speed of a particular point and equals the angular velocity multiplied by the perpendicular distance from the axis. On the Earth that distance is the radius of the circle of latitude, which is largest at the equator and shrinks to nothing at the poles. Hence one common angular velocity and a linear velocity that varies with the cosine of the latitude.
The mental picture that settles this is a spinning merry-go-round. Everyone on it completes a circle in the same time, so the angular rate is shared, but a child on the outer rail travels much farther in that time than one near the centre. Extend it to the Earth and the equator is the outer rail, the poles the centre. This is also why rockets are launched eastward from sites near the equator — the launch pad is already moving at close to 1,675 kilometres an hour, and that speed comes free.
- The Earth's angular speed of rotation is about 7.2921 by ten to the power minus five radians per second, the same for every point on the surface.
- Linear speed at the equator is about 1,674.7 kilometres an hour, roughly 465 metres per second.
- Linear speed at any latitude equals the equatorial speed multiplied by the cosine of that latitude — about 1,184 kilometres an hour at 45 degrees.
- At the poles the linear speed of rotation is effectively zero, because a point there covers no distance in a full turn.
Linear speed equals the equatorial speed times the cosine of the latitude; angular speed does not vary at all.
- Assuming that because the equator moves fastest, it must also turn through a bigger angle — the angle is the same everywhere.
- Believing we should feel the Earth's speed; the motion is uniform, the atmosphere turns with us, and there is no nearby reference object.
- Mixing up the cosine rule for linear speed with the sine rule for the Coriolis parameter, which behaves the opposite way with latitude.
Asked as a two-statement item testing whether the candidate can separate angular from linear velocity on a rotating Earth.
Inhabitants are unaware of the speed of rotation of the planet Earth because 1. the angular velocity is constant for each place on the Earth’s surface 2. the atmosphere rotates with the Earth 3. there are no nearby objects, either stationary or moving at a rate different from that of the Earth Which of the above is/are the correct explanation(s)?
- (a) 1 only
- (b) 1 and 2 only
- (c) 2 and 3 only
- (d) 1, 2 and 3
Answer(d) 1, 2 and 3
Statement 1 there is statement 1 here, reused as an explanation rather than as a fact to verify. That NDA item takes the constancy of angular velocity and asks what follows from it for our everyday experience, which is a good check on whether you have really understood the idea.
- practice — not a real PYQ
The linear speed due to the Earth's rotation is greatest at
- (a)the North Pole
- (b)the Arctic Circle
- (c)the equator
- (d)the Tropic of Cancer
Answer(c) the equator — it traces the largest circle of latitude, so it covers the greatest distance in one rotation.
- practice — not a real PYQ
Rocket launch sites are often located close to the equator mainly because
- (a)gravity is much weaker there
- (b)the launch pad already moves eastward at nearly 1,675 kilometres an hour
- (c)the atmosphere is thinner over the equator
- (d)the Coriolis force assists the ascent
Answer(b) the launch pad already moves eastward at nearly 1,675 kilometres an hour — an eastward launch inherits that speed.