The Earth takes slightly less than one day to complete one rotation about the axis passing through its poles. The actual time is `8.616xx10^(4)`s. With this given information, what is the angular speed of the Earth about its axis?

Options

A. `7.292xx10^(-5)` rad/s
B. `9.951xx10^(-5)` rad/s
C. `2.321xx10^(-6)` rad/s
D. `6.334xx10^(-4)` rad/s

Answer

To find the angular speed of the Earth about its axis, we can follow these steps:

### Step 1: Understand the relationship between angular speed and time period
The angular speed (ω) of an object in circular motion is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
where \(T\) is the time period for one complete rotation.

### Step 2: Identify the time period
From the problem statement, the time period \(T\) for one complete rotation of the Earth is given as:
\[ T = 8.616 \times 10^4 \text{ seconds} \]

### Step 3: Substitute the time period into the angular speed formula
Now, we can substitute the value of \(T\) into the formula for angular speed:
\[ \omega = \frac{2\pi}{8.616 \times 10^4} \]

### Step 4: Calculate the angular speed
First, we calculate \(2\pi\):
\[ 2\pi \approx 6.2832 \]
Now, substituting this value into the equation:
\[ \omega = \frac{6.2832}{8.616 \times 10^4} \]

### Step 5: Perform the division
Calculating the division:
\[ \omega \approx \frac{6.2832}{86160} \approx 0.0000728 \text{ radians/second} \]

### Step 6: Express the result in scientific notation
To express \(0.0000728\) in scientific notation:
\[ \omega \approx 7.28 \times 10^{-5} \text{ radians/second} \]

### Final Result
Thus, the angular speed of the Earth about its axis is:
\[ \omega \approx 7.28 \times 10^{-5} \text{ radians/second} \]

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