GATE 2023 CE (CE2) – Question 10
Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively. Which one of the following options is CORRECT?
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Correct answer: (C) $v_P>v_R>v_Q$
Explanation
All points on the globe turn with the same angular speed $\omega$ about the polar axis, but each moves on a circle whose radius depends on its latitude $\lambda$:
$$r=R\cos\lambda,\qquad v=\omega r=\omega R\cos\lambda .$$
- **P (equator, $\lambda=0^\circ$):** $v_P=\omega R$, the largest.
- **R (midway, $\lambda=45^\circ$):** $v_R=\omega R\cos45^\circ=\dfrac{\omega R}{\sqrt2}\approx0.71\,\omega R$.
- **Q (north pole, $\lambda=90^\circ$):** $v_Q=\omega R\cos90^\circ=0$ (it only spins on the spot).
So $v_P>v_R>v_Q$, option **C**.