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 theta=0 rad at noon on Monday and moving in the positive theta direction, what is your position theta in rad on Tuesday at 11:00 am?
To answer this question, we need to find the elapsed time from Monday at noon to Tuesday at 11:00 am.
On Monday at noon, we start at theta = 0 rad.
From Monday at noon to Tuesday at noon, 24 hours have passed.
From Tuesday at noon to Tuesday at 11:00 am, 11 hours have passed.
So, the total elapsed time is 24 hours + 11 hours = 35 hours.
To find the position theta in rad after 35 hours, we can use the formula:
theta = (time * 2pi) / period
where time is the elapsed time in seconds and the period is the length of one full rotation in seconds.
In this case, the period is 24 hours * 60 minutes * 60 seconds = 86,400 seconds.
Converting the elapsed time to seconds:
35 hours * 60 minutes * 60 seconds = 126,000 seconds.
Plugging in these values into the formula:
theta = (126,000 seconds * 2pi) / 86,400 seconds
theta = 3.665 rad
Therefore, your position theta in rad on Tuesday at 11:00 am would be approximately 3.665 rad.