Problem 1C-05*: Estimate the Diameter of the Moon

A simple experiment can measure the diameter of the moon if the distance from the earth to the moon, L, is given. On a clear night lit by the full moon, plant a stick into the ground and align your eye with the tip of the stick and a point P on the edge of the moon as shown in part (a) of the figure. Start to measure time by activating a stopwatch. Read the time elapsed when point Q, on the other side of the edge of the moon, aligns with the tip of the stick and your eye as shown in part (b) of the figure. Let the elapsed time be denoted as T. From T and L, estimate the diameter of the moon, D.

Note: Points P and Q should be chosen to make PQ closely approximate a diameter of the moon. You may need to pratice for a while to determine an appropriate PQ pair.

Warning: Do not attempt to perform this experiment with the sun; it will damage your eye when you look directly into the sun. Even for the moon it is advisable to wear sunglasses to protect your eyes.

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Solution:

What we want to measure is the tiny angle θ that span the diameter of the moon as shown in the second figure. Let θ be expressed in radians. Then

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The movement of the moon in the night sky is due to the rotation of the earth. After 24 hours the moon will return roughly to the same spot on the sky. In other words in 24 hours the moon will go around the sky 360 degrees, or 2π radians. Since 24 hours = 24×60×60 = 8.64×104 seconds, we get
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On the other hand if we know the value of D, that is 3.48×103 km, and use L = 3.84×105 km, then we can estimate T to be around 125 seconds.