What is the angular velocity of the earth?
(A) \[\dfrac{{2\pi }}{{86400}}rad/\sec \]
(B) \[\dfrac{{2\pi }}{{3600}}rad/\sec \]
(C) \[\dfrac{{2\pi }}{{24}}rad/\sec \]
(D) \[\dfrac{{2\pi }}{{6400}}rad/\sec \]

Hint: Angular velocity of the earth is the velocity of the earth which it takes to complete one revolution around the sun. Generally angular velocity is given as the ratio of \[2\pi \]and the time period taken by the particle to complete one revolution. Using this find the angular velocity of the earth.
Formula used:
\[\omega = \dfrac{{2\pi }}{T}\]

Complete step by step solution:
Angular speed of a body is defined as the rate of change of angular displacement per unit time . It is a vector quantity which expresses both direction and magnitude. Now for one revolution, the total angle covered by the body is\[2\pi \].
Earth is the 3rd planet in the solar system that rotates around the sun. The time taken by earth to spin around its orbit is 365 days. So, earth displaces itself at an angle, in 24 hours or a day. As days progress, earth spins around its orbit and on the 365th day, it comes closer to the sun.
Now, we need to calculate the time period earth takes to complete one full rotation around it’s geographical axes. According to Hipparchus, earth completes one full rotation in 24 equinoctial hours, which is purely based on sunlight and darkness calculation. Based on this , we can find the angular velocity, which is given as
\[ \Rightarrow \omega = \dfrac{{2\pi }}{T}\]
Now, \[T\]is 24 hours. We can write this in seconds as ,
\[ \Rightarrow \omega = \dfrac{{2\pi }}{{24 \times 60 \times 60s}}\]
On further simplifying, we obtain
\[ \Rightarrow \omega = \dfrac{{2\pi }}{{24 \times 36 \times 100s}}\]
\[ \Rightarrow \omega = \dfrac{{2\pi }}{{86400}}rad/\sec \]
Thus the earth moves in an angular velocity of \[\dfrac{{2\pi }}{{86400}}rad/\sec \]per day.

Hence option (a) is the right answer.

Note: Initially the clocking system was based upon the sunrise and sunset period. Sundials were used in the olden day civilizations to identify time that uses the sun's shadow marked against a large dial. Angular velocity of earth in a year can be calculated as earth’s angular velocity in a day multiplied by 365 days.