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 situated at different locations on Earth, we need to use the formula for centripetal acceleration:
Ac = ω^2 * r
Where:
- Ac is the centripetal acceleration
- ω is the angular velocity
- r is the distance from the axis of rotation
In this case, the Earth's axis of rotation passes through the North and South Poles, and the distance from the axis of rotation will vary depending on the location on Earth.
(a) At the Equator:
At the equator, the distance from the axis of rotation is the Earth's radius, r = 6.38 x 10^6 m. The angular velocity, ω, can be calculated using the period of rotation, T, which is 24 hours or 86,400 seconds:
ω = 2π / T
Substituting the value for T, we get:
ω = 2π / 86,400
Now, we can calculate the centripetal acceleration:
Ac = ω^2 * r
= (2π / 86,400)^2 * 6.38 x 10^6
(b) At a Latitude of 15.0° North:
To calculate the centripetal acceleration at a specific latitude, we need to consider the distance from the axis of rotation. The distance from the axis of rotation at a given latitude can be calculated using the formula:
r = R * cos(latitude)
Where R is the Earth's radius and latitude is the specific latitude.
Substituting the values, we get:
r = (6.38 x 10^6) * cos(15.0°)
Now, we can calculate the centripetal acceleration using the formula:
Ac = ω^2 * r
= (2π / 86,400)^2 * ((6.38 x 10^6) * cos(15.0°))
Now you can plug in these values into a scientific calculator or computer program to calculate the centripetal acceleration for both situations.