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October 20th, 2007, 10:54 AM   #1
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Degrees and Radians

The earth rotates on its axis once every 24 hours.
a)How long does it take earth to rotate through an angle of 4pi/3?
b)how long does it take earth to rotate through an angle of 120 degree?
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October 20th, 2007, 11:32 AM   #2
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There are 360 degrees and 2*pi radians in a complete revolution. Does that help? Just find the ratio between the angle being traveled by the Earth in the problem and the number of degrees/radians in a full revolution, and then multiply that by the amount of time it takes the Earth to rotate around its axis.
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October 20th, 2007, 11:34 AM   #3
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The first thing to know about radians and degrees is that there are 2π radians and 360 degrees in a circle. That is, one complete revolution (rotation) is 2π radians, or 360 degrees.

Some angle measures you should get used to are (you can just glance over these for now... you'll see them enough in your life)
30º or π/6, which is 1/12 of a circle (Think of a clock)
45º or π/4 which is 1/8 of a circle (Cut a square in half diagonally)
60º or π/3 which is 1/6 of a circle (The compliment of 30º, or 2 hours)
90º or π/2 (A right angle.)
120º or 2π/3 (same distance past 90º as 60º is from 90º)
180º or π

After half a circle, it's all the same as that, plus π (or 180º)

Moving on, it takes 24 hours to rotate 2π radians (360º)
For good practice, I'll do both problems in radians and degrees:

h/24= (4π/3)/(2π) <-- It takes 24 hours to go a complete circle, so how many hours (h) does it take to go 4π/3 radians?

First, get rid of π on both sides:
h/24 = (4/3)/2 -> h = 24•2/3 -> h = 16 <--There's the answer to part one

Also (in degrees),
There are 360º/2π, or 180º/π so there are x=[(4π/3) • (180/π)]º in 4π/3
4π/3 • (180º/π) = 240º

240/360 = h/24
2/3 = h/24
h = 16.
As long as we didn't screw up the conversion, the answers should be the same using either radians or degrees.

Same thing with #2:

120/360 = h/24
1/3 = h/24
h = 8

120•(π/180) = 2π/3
(2π/3)/(2/π) = h/24
1/3 = h/24
h = 8.
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October 21st, 2007, 02:47 PM   #4
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Slight nitpicking. The earth takes slightly less (I think about 4 min.) than 24 hours to rotate 360 deg. The extra time is needed to complete an earth day because the earth has moved 1/365th of its distance in orbit around the sun.
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October 21st, 2007, 04:32 PM   #5
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I can't see how motion around the sun influences the length of the day. If we define a point that the Earth starts from and ends from in regards to spinning motion, moving around the Sun shouldn't influence that (I think).
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October 21st, 2007, 06:25 PM   #6
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Quote:
Originally Posted by Infinity
I can't see how motion around the sun influences the length of the day.
If you had permanent day, the rotation rate would be one rotation per year, wouldn't it?
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October 21st, 2007, 07:08 PM   #7
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Quote:
I can't see how motion around the sun influences the length of the day.
OK, so you are defining the day as how much time we get sunlight. I am defining it as how long it takes the Earth to spin once (360 degrees) on its axis.

Quote:
If you had permanent day, the rotation rate would be one rotation per year, wouldn't it?
Yes, similar to the moon going around the Earth.
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October 22nd, 2007, 01:48 PM   #8
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Quote:
Originally Posted by Infinity
I can't see how motion around the sun influences the length of the day. If we define a point that the Earth starts from and ends from in regards to spinning motion, moving around the Sun shouldn't influence that (I think).
A day is defined in terms of how the earth is facing the sun, that is from noon to noon, where noon is defined as the time that the sun is directly overhead (at the appropriate latitude). Since the earth has moved in its orbit, it has to rotate more than 360 deg. to get there.
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