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October 20th, 2007, 10:54 AM  #1 
Newbie Joined: Oct 2007 Posts: 19 Thanks: 0  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? 
October 20th, 2007, 11:32 AM  #2 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
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.

October 20th, 2007, 11:34 AM  #3 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
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. 
October 21st, 2007, 02:47 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,607 Thanks: 616 
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.

October 21st, 2007, 04:32 PM  #5 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
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).

October 21st, 2007, 06:25 PM  #6  
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October 21st, 2007, 07:08 PM  #7  
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October 22nd, 2007, 01:48 PM  #8  
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