|Sep22-10, 06:35 PM||#1|
**Quick angular measurement question about moon. Thanks!
I am suppose to write a calculation on finding angular measurement of the moon. Does the following make sense, and is correct?
|Sep22-10, 09:01 PM||#2|
Let me refresh your memory on how we measure angles. You can imagine drawing a radial line straight from the observer to one end of the object (in this case the moon). Then you can imagine drawing another radial line from the observer to the other end of the object. Going from one line to another, you sweep out a circular arc (a portion of a circle), since the two radii have the same length. Let's call the length of this circular arc 's', and the radial distance 'r.' The definition of the angle θ between the two lines is θ = s/r. When defined in this way, angles are measured in units of radians, which are dimensionless units (since the angles are defined as a ratio of two lengths). I've drawn a diagram to help illustrate what I mean. NOTE: using the symbols you used, s = do and r = d.
In the diagram, the two radial lines and the circular arc they cover make up a sort of pie-shaped wedge. Now, clearly, to cover an angle of 1 radian, the arc length s would have to be equal to the radius r. In that case, the pie-shaped wedge would be very close in shape to an equilateral triangle (since two sides have length r, and the third "side", which is curved, also has length r). Therefore, you would expect θ to be close to 60 degrees, but not quite. As it happens, θ = 57.3 degrees. So 1 radian = 57.3 degrees. THAT's where the 57.3 comes from. You don't need any of this nonsense about rulers and whatnot.
Since my argument about an equilateral triangle doesn't give an exact answer, I should point out that the way to get the exact conversion factor is as follows: when you sweep out a full circle, this corresponds to an angle of 2π radians. Hence, 2π radians = 360 degrees, or
1 radian = 360 degrees / 2π = 57.3 degrees.
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