Calculate the acceleration due to the rotation of the Earth of a person in a location 26° northern geographical latitude. Use REarth = 3,686 km for the value of the radius of the Earth.
To calculate the acceleration due to the rotation of the Earth at a given latitude, we can use the formula:
a = R * ω^2 * cos(φ)
where:
- a is the acceleration due to rotation,
- R is the radius of the Earth,
- ω is the angular velocity of the Earth's rotation, and
- φ is the latitude.
First, we need to calculate the angular velocity, ω. The angular velocity can be given by:
ω = (2π) / T
where T is the period of the Earth's rotation. The period of rotation is approximately 24 hours or 86,400 seconds.
Plugging in the values, we have:
ω = 2π / 86,400
Now, let's calculate the angular velocity:
ω ≈ 7.27 × 10^-5 rad/s
Next, we can calculate the acceleration due to rotation at the given latitude. Plugging in the values of R and φ:
a = 3,686 km * (7.27 × 10^-5 rad/s)^2 * cos(26°)
To calculate in SI units, we need to convert the radius of the Earth from kilometers to meters. 1 km is equal to 1000 meters, so:
R = 3,686 km * 1000 = 3,686,000 meters
Now we can substitute the values and calculate the acceleration due to the rotation:
a = 3,686,000 m * (7.27 × 10^-5 rad/s)^2 * cos(26°)
Using a calculator, we get the result:
a ≈ 3.38 × 10^-2 m/s^2
Therefore, the acceleration due to the rotation of the Earth at a latitude of 26° is approximately 3.38 × 10^-2 m/s^2.