As denizens of the surface of a spinning planet, we are always in uniform circular motion. Imagine you are in Nairobi (on the Earth's equator) at noon on a Monday. Answer the following questions only considering the rotation of the earth and NOT the Earth's circular motion around the sun. The radius of the earth is 6371 km. A day is 24 hours.
Assuming that you are at at theta = 0 rad at noon on Monday and moving in the positive direction, what is your position theta in rad on Tuesday at 11:00 am?
I'm assuming this has something to do with orbit and rotation, but what exactly should I be using?
Well, if I'm understanding correctly, you want to find your position on Tuesday at 11:00 am, given that you started at noon on Monday at Nairobi's equator. Since a day is 24 hours, the time from noon on Monday to 11:00 am on Tuesday is 23 hours.
Now, considering only the rotation of the Earth, we can calculate the angle covered in that time. The Earth completes one full rotation in 24 hours, so in 23 hours, you will be short of completing a full rotation by 1/24th.
Since a full rotation is 2π radians, we can calculate your position theta by multiplying 2π by the fraction of the day you have completed. In this case, it will be 2π * (23/24).
So, your position theta in radians on Tuesday at 11:00 am would be approximately 5.76 radians. Just keep in mind that this calculation only considers the Earth's rotation and not its orbit around the Sun.
To determine your position theta in radians on Tuesday at 11:00 am, we need to consider the rotation of the Earth.
First, let's calculate the time difference between noon on Monday and 11:00 am on Tuesday. There are 24 hours in a day, so the time difference is 23 hours.
Next, let's calculate the angular displacement you would have experienced during this time. The Earth completes one full rotation in 24 hours, which corresponds to 2π radians. Therefore, the angular displacement per hour is:
Angular displacement per hour = (2π radians) / (24 hours) = π/12 radians per hour
Now, we can calculate the total angular displacement from noon on Monday to 11:00 am on Tuesday:
Total angular displacement = (π/12 radians per hour) x (23 hours) = 23π/12 radians
Finally, we need to add this total angular displacement to your original position at noon on Monday:
Position theta on Tuesday at 11:00 am = 0 radians + 23π/12 radians
So, your position theta on Tuesday at 11:00 am is 23π/12 radians.
To find your position theta on Tuesday at 11:00 am, considering only the rotation of the Earth, you can use the concept of angular displacement and the formula for calculating it.
Since a day is 24 hours, and you are at theta = 0 rad at noon on Monday, you have already completed half a rotation (180 degrees) by Monday at 12:00 am. So, you can start by calculating the angular displacement for the remaining time from Monday at 12:00 am to Tuesday at 11:00 am.
The formula for angular displacement is given by:
θ = ω * t
Where:
θ is the angular displacement (in radians)
ω is the angular velocity (in radians per second)
t is the time (in seconds)
To calculate ω, we need to convert the time into seconds. From Monday at 12:00 am to Tuesday at 11:00 am, there are 23 hours since there are 24 hours in a full day. Therefore, the remaining time is 23 hours or 23 * 60 * 60 seconds.
The angular velocity ω can be calculated as the distance traveled divided by the time taken. The distance traveled is the circumference of the Earth, 2πr, where r is the radius of the Earth.
ω = (2πr) / t
Substituting the values, we get:
ω = (2π * 6371 km) / (23 * 60 * 60 s)
Now, using the calculated value of ω, we can find the angular displacement θ:
θ = ω * t
Substituting the time from Monday at 12:00 am to Tuesday at 11:00 am (23 hours or 23 * 60 * 60 seconds), we can calculate θ.
Note: Make sure to convert the radius of the Earth into the same unit as the distance traveled.
It's important to remember that this calculation only considers the rotation of the Earth and not its circular motion around the Sun.
Monday at noon to Tuesday at 11 am is 23 hours.
At Noon in Nairobi the sun is overhead on the equator (assuming on a solstice day when the sun is over the equator)
The sun stays there, but I move east with the earth rotation and 23 hours later I have gone 23/24 of the way around
(23/24)360 = 345 degrees
In another hour or 15 degrees it will be overhead again.