To calculate the change in the value of `g` at latitude `45°`, we can use the formula that accounts for the effect of the Earth's rotation. The formula for the change in acceleration due to gravity `g'` at latitude `θ` is given by:
\[ g' = g - r \omega^2 \cos^2(\theta) \]
Where:
- \( g \) is the acceleration due to gravity at the equator (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( r \) is the radius of the Earth,
- \( \omega \) is the angular velocity of the Earth,
- \( \theta \) is the latitude.
### Step-by-Step Solution:
1. **Convert the Radius of the Earth:**
Given \( R = 6.37 \times 10^3 \, \text{km} = 6.37 \times 10^6 \, \text{m} \).
2. **Calculate the Angular Velocity (\( \omega \)):**
The angular velocity of the Earth can be calculated using the formula:
\[
\omega = \frac{2\pi}{T}
\]
where \( T \) is the time period of rotation in seconds. For one complete rotation (24 hours):
\[
T = 24 \times 60 \times 60 = 86400 \, \text{s}
\]
Thus,
\[
\omega = \frac{2\pi}{86400} \approx 7.27 \times 10^{-5} \, \text{rad/s}
\]
3. **Calculate \( \cos^2(45°) \):**
Since \( \cos(45°) = \frac{1}{\sqrt{2}} \),
\[
\cos^2(45°) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}
\]
4. **Substitute Values into the Formula:**
Now, substituting the values into the formula for the change in `g`:
\[
g' - g = - R \omega^2 \cos^2(45°)
\]
\[
g' - 9.81 = - (6.37 \times 10^6) \times (7.27 \times 10^{-5})^2 \times \frac{1}{2}
\]
5. **Calculate \( R \omega^2 \):**
First, calculate \( \omega^2 \):
\[
\omega^2 = (7.27 \times 10^{-5})^2 \approx 5.29 \times 10^{-9} \, \text{(rad/s)}^2
\]
Now, calculate \( R \omega^2 \):
\[
R \omega^2 = (6.37 \times 10^6) \times (5.29 \times 10^{-9}) \approx 0.0337 \, \text{m/s}^2
\]
6. **Calculate the Change in `g`:**
Now substitute back into the equation:
\[
g' - 9.81 = -0.0337 \times \frac{1}{2}
\]
\[
g' - 9.81 = -0.01685
\]
\[
g' = 9.81 - 0.01685 \approx 9.79315 \, \text{m/s}^2
\]
7. **Final Change in `g`:**
The change in the value of `g` at latitude `45°` is:
\[
\Delta g = g' - g = -0.01685 \, \text{m/s}^2 \approx -0.017 \, \text{m/s}^2
\]
### Summary:
The change in the value of `g` at latitude `45°` is approximately \( -0.017 \, \text{m/s}^2 \).
