What is the angular velocity of the earth?

Options

A. `(2pi)/86400rad//s`
B. `(2pi)/3600rad//s`
C. `(2pi)/24rad//s`
D. `(2pi)/6400rad//s`

Answer

To find the angular velocity of the Earth, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Concept of Angular Velocity**:
Angular velocity (ω) is defined as the rate of change of angular displacement and is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period of one complete rotation.

2. **Determine the Time Period (T)**:
The Earth takes approximately 24 hours to complete one rotation about its axis. This is our time period \( T \).

3. **Convert Time Period to Seconds**:
Since angular velocity is typically expressed in radians per second, we need to convert 24 hours into seconds:
\[ T = 24 \text{ hours} \times 3600 \text{ seconds/hour} = 86400 \text{ seconds} \]

4. **Substitute the Time Period into the Angular Velocity Formula**:
Now we can substitute \( T \) into the angular velocity formula:
\[ \omega = \frac{2\pi}{86400 \text{ seconds}} \]

5. **Calculate the Angular Velocity**:
Now we perform the calculation:
\[ \omega = \frac{2\pi}{86400} \approx 7.272 \times 10^{-5} \text{ radians/second} \]

### Final Answer:
The angular velocity of the Earth is approximately \( 7.272 \times 10^{-5} \) radians per second.

---