Question

What is the angular velocity of the Earth?
A. $\dfrac{2\pi }{86400}\text{rad/s}$
B. $\dfrac{2\pi }{3600}\text{rad/s}$
C. $\dfrac{2\pi }{24}\text{rad/s}$
D. $\dfrac{2\pi }{6400}\text{rad/s}$

Answer 1

Hint: Angular velocity is the velocity associated with a particle executing a rotational motion. It can be defined as the rate of change of angular position with respect to an origin. We know that Earth takes 24 hours to complete one rotation on its axis. An object has to cover an angle of $2\pi$ in order to complete one complete rotation.

Equation Used:
The angular velocity of an object is defined as,
$\omega =\dfrac{2\pi }{T}$
Where,
$\omega$ is the angular velocity.
$\text{T}$ is the time period to complete one rotation.

Complete step by step answer:
The angular velocity can be defined as the rate of change of angular position with respect to an origin. So, it can be defined mathematically as,
$\omega =\dfrac{2\pi }{T}$
Where,
$\omega$ is the angular velocity.
$\text{T}$ is the time period to complete one rotation.
So, in the case of Earth, the time period to complete one rotation is 24 hours. Which can be expressed as $24\times 60\times 60$ seconds. So, the angular velocity of the Earth can be expressed as,
${{\omega }_{e}}=\dfrac{2\pi }{{{T}_{e}}}$
${{\omega }_{e}}$ is the angular velocity of the Earth.
${{\text{T}}_{e}}$ is the time period of the Earth to complete one rotation.
${{\omega }_{e}}=\dfrac{2\pi }{24\times 60\times 60}$
${{\omega }_{e}}=\dfrac{2\pi }{86400}$
So, the answer to the question is the option (A).

Note: The SI unit of angular velocity is radians per second $\left( \text{rad}/\text{second} \right)$.
The linear velocity of a particle executing rotational motion about an origin at a distance r away from the origin can be expressed as the cross product of the angular velocity of the body and the radius of the orbit.
In a geostationary satellite which are communication satellites, the time period of the satellite is equal to the time period of the Earth. It is focused at a fixed point on the Earth and rotates along with that fixed point. So, the angular velocity of these satellites will be equal to that of the Earth.