The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator. Assuming the earth is a sphere with a radius of 6.38 x 106 m, determine the centripetal acceleration of a person situated (a) at the equator and (b) at a latitude of 15.0 ° north of the equator
To determine the centripetal acceleration of a person at different locations on Earth, we need to first understand the concept of centripetal acceleration. Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. In this case, the circular path is the rotation of the Earth.
The centripetal acceleration can be calculated using the formula:
a = (v^2) / r
Where:
a = centripetal acceleration
v = linear velocity of the object
r = radius of the circular path
To calculate the centripetal acceleration at the equator (a), we need to determine the linear velocity (v) and the radius (r) of the circular path.
To calculate the linear velocity (v), we can use the formula:
v = 2πr / T
Where:
v = linear velocity
π ≈ 3.14 (pi)
r = radius of Earth
T = period of rotation (which is equal to one day)
Substituting the given values, we have:
v = 2π(6.38 x 10^6 m) / (24 x 60 x 60 s)
Simplifying this equation, we find:
v ≈ 464.1 m/s
Now that we have the value for the linear velocity (v), we can calculate the centripetal acceleration (a) using the formula:
a = (v^2) / r
Substituting the values:
a = (464.1 m/s)^2 / (6.38 x 10^6 m)
Simplifying this equation, we find:
a ≈ 0.034 m/s^2
Therefore, the centripetal acceleration at the equator is approximately 0.034 m/s^2.
To calculate the centripetal acceleration at a latitude of 15.0° north of the equator (b), we need to adjust the radius (r) accordingly. Since the Earth is not a perfect sphere but slightly flattened at the poles, the radius at different latitudes differs slightly.
The radius at a latitude of 15.0° north of the equator can be calculated using the formula:
r = cos(lat) x radius of Earth
Where:
r = adjusted radius at latitude
lat = latitude in radians (15.0° converted to radians)
radius of Earth = 6.38 x 10^6 m
Substituting the given values, we have:
r = cos(15°) x (6.38 x 10^6 m)
Simplifying this equation, we find:
r ≈ 6.32 x 10^6 m
Now, using the same formula for centripetal acceleration:
a = (v^2) / r
Substituting the values, we have:
a = (464.1 m/s)^2 / (6.32 x 10^6 m)
Simplifying this equation, we find:
a ≈ 0.035 m/s^2
Therefore, the centripetal acceleration at a latitude of 15.0° north of the equator is approximately 0.035 m/s^2.