Question

The angular velocity of earth about its axis of rotation is:
$(a){\text{ }}\dfrac{{2\pi }}{{\left( {60 \times 60 \times 24} \right)}}rad/\sec \\\ (b){\text{ }}\dfrac{{2\pi }}{{\left( {60 \times 60} \right)}}rad/\sec \\\ (c){\text{ }}\dfrac{{2\pi }}{{60}}rad/\sec \\\ (d){\text{ }}\dfrac{{2\pi }}{{\left( {365 \times 24 \times 60 \times 60} \right)}}rad/\sec \\\$

Answer 1

- Hint – In this question use the concept that angular velocity $\omega = \dfrac{{\Delta \theta }}{{\Delta t}}$, where $\Delta \theta = 2\pi$ and the time taken by earth to complete one complete rotation is 1 day that is 24 hours.

Complete step-by-step solution -

The angular velocity ($\omega$) of earth is defined as one complete revolution in 1 day.
$\Rightarrow \omega = \dfrac{{\Delta \theta }}{{\Delta t}}$................. (1), where $\Delta \theta$ = one complete revolution and $\Delta t$ = 1 day.
Now as we know in a complete revolution there is ${360^0}$ (or) $2\pi$radians.
$\Rightarrow \Delta \theta = 2\pi$ Radians.
And in 1 day there are 24 hours.
And in 1 hour there is 60 minutes.
So in 24 hours there is $\left( {24 \times 60} \right)$ minutes.
Now in 1 minute there is 60 sec.
So in $\left( {24 \times 60} \right)$ minutes there is $\left( {24 \times 60 \times 60} \right)$ sec.
$\Rightarrow \Delta t = 1{\text{ day}} = \left( {24 \times 60 \times 60} \right)$ Seconds.
Now substitute the values in equation (1) we have,
$\Rightarrow \omega = \dfrac{{\Delta \theta }}{{\Delta t}} = \dfrac{{2\pi }}{{24 \times 60 \times 60}}$ rad/sec.
So this is the required answer.
Hence option (A) is correct.

Note – In this question the options were having the specific units of $rad/\sec$ that’s why the complete rotation angle was taken in radians and not in degrees moreover 24 hours were converted into seconds. This is not standardized and measurement of units can vary from question to question. There is confusion between rotation and revolution, earth along with other planets revolves around the sun, however it revolves around its own axis. The revolution around the sun takes 365 days and it forms a year while one rotation takes 24 hours and forms one complete day.