How long would it take for the Earth to complete a full turn if a person at 49.2° northern geographical latitude floats apparently weightlessly across the room? Use REarth = 6,385 km for the radius of Earth
How long would it take for the Earth to complete a full turn if a person at 49.2° northern geographical latitude floats apparently weightlessly across the room? Use REarth = 6,385 km for the radius of Earth.
Not sure if I understood your question.
I interpret it as saying "if the Earth is rotating at a yet unknown angular velocity ω such that a person would float weightlessly at latitude 49.2°N, find &omega."
It is not as simple as it sounds, because the acceleration due to gravity acts towards the centre of the Earth. On the other hand, the rotation of the Earth is around a N-S axis, causing the centripetal force to be at an angle θ with the vertical, where θ is the latitude.
Assuming that the vertical (towards the centre of the earth) components balance, and the person floating is restrained from flying south by a horizontal rope, then we can do the following calculations:
Acceleration due to gravity, g = 9.8 m/s²
Radius of the Earth, R = 6385 km = 6385000 m
Latitude = 49.2°
We will find r, the distance of the surface of the earth to the axis of rotation, i.e. measured along the equatorial plane.
r = Rcos(θ)
Centripetal acceleration, a
= rω² (perpendicular to axis of rotation)
Vertical component of centripetal acceleration, av
= a cos(θ)
= rω² cos(&theta)
= Rω&up2; cos2(&theta)
Equate av and g, solve for &omega.
I get 0.0019 radians/sec. which translats to a full rotation in 55 minutes and 14 seconds.
Editorial correction:
Vertical component of centripetal acceleration, av
= a cos(θ)
= rω² cos(θ)
= R ω² cos²(θ)
Here's an article complete with figures for supplementary reading:
http://galitzin.mines.edu/INTROGP/notes_template.jsp?url=GRAV%2FNOTES%2Flatitude.html&page=Gravity%3A%20Notes%3A%20Latitude%20Variations
how did you translate 0.0019 radians/sec into minutes?
To determine how long it would take for the Earth to complete a full turn while a person at a specific latitude floats weightlessly, we need to consider the concept of Earth's rotation and the definition of a full turn.
The Earth completes one full rotation about its axis in approximately 24 hours, which is equivalent to one day. This means that every place on Earth will experience one full turn in this time frame.
However, the rotation of the Earth is not the same everywhere. At different latitudes, the rotational speed of the Earth varies. The rotational speed is highest at the equator and decreases as you move towards the poles.
To calculate the time it takes for one full turn at a specific latitude, we need to adjust for the distance traveled due to the rotation of the Earth. This can be done using the formula:
Time = Distance / Speed
In this case, the distance we are interested in is the circumference of the Earth at the given latitude, and the speed is the rotational speed of the Earth at that latitude.
To calculate the circumference of the Earth at a specific latitude, we can use the formula for the circumference of a circle:
Circumference = 2 * π * Radius
Given that the radius of the Earth (REarth) is 6,385 km, we can calculate the circumference at the given latitude by substituting the radius into the formula:
Circumference = 2 * π * 6,385 km
Next, we need to calculate the rotational speed of the Earth at the given latitude. The rotational speed can be calculated using the equation:
Speed = (2 * π * Radius) / Time
Since we know that the time for one full turn is 24 hours, we can calculate the rotational speed by substituting the values into the equation:
Speed = (2 * π * 6,385 km) / 24 hours
Now, we can calculate the time it would take for the Earth to complete one full turn at the given latitude by dividing the circumference by the rotational speed:
Time = Circumference / Speed
By substituting the values we have calculated into this equation, we can find the answer to the question. Remember to convert the units appropriately to get the answer in the desired format.
It's worth noting that although this calculation provides an estimate, the actual time it takes for the Earth to complete a full turn at a specific latitude may be slightly different due to factors such as irregularities in the Earth's shape, gravitational effects, and other atmospheric conditions.