It takes 23 hours 56 minutes and 4 seconds for the earth to make one revolution (mean sidereal day). What is th 7.29×10-5 rad/s
The Correct Answer and Explanation is :
The angular velocity of the Earth can be calculated using the formula for angular velocity, which is:
[ \omega = \frac{\theta}{t} ]
where:
- ( \omega ) is the angular velocity in radians per second,
- ( \theta ) is the angular displacement in radians,
- ( t ) is the time period in seconds.
For one complete revolution of the Earth, the angular displacement ( \theta ) is ( 2\pi ) radians (since a full circle is ( 2\pi ) radians). The time period ( t ) for one mean sidereal day is 23 hours, 56 minutes, and 4 seconds. Converting this time into seconds:
[ t = (23 \times 3600) + (56 \times 60) + 4 = 86164 \text{ seconds} ]
Now, substituting the values into the formula gives:
[ \omega = \frac{2\pi}{86164} ]
[ \omega \approx \frac{6.28319}{86164} \approx 7.29 \times 10^{-5} \text{ rad/s} ]
This calculation shows that the Earth rotates at an angular velocity of approximately ( 7.29 \times 10^{-5} \text{ rad/s} ) in a mean sidereal day. This value represents how quickly the Earth rotates about its axis in relation to fixed stars, rather than the Sun. A sidereal day is used in astronomy to track the position of stars and other celestial objects, and it differs from the solar day (24 hours) because of the Earth’s simultaneous orbit around the Sun.
Understanding the angular velocity in this context helps us appreciate the precision required in astronomical observations and calculations, which rely on knowing the exact movement of Earth to accurately position telescopes and predict celestial events.
To visualize this concept, let’s create an image showing the Earth in space, with an overlay of a radial line representing one complete rotation to illustrate the concept of a sidereal day and angular velocity.
Here’s the image visualizing the Earth in space with a radial line marking one complete rotation. This representation helps illustrate the concept of Earth’s angular velocity during a sidereal day.
