## Sunday, October 9, 2011

### Measuring the Radius of the Earth

In this lab, we're going to calculate the radius of the Earth using nothing but a stopwatch and the horizon. We chose the lovely Santa Monica Beach for our experiment. As the sun began approaching the horizon, I lay down to be eye level with the ocean. As soon as I saw the bottom of the sun touch the horizon, I gave the signal, and my partners started their stopwatches. As soon as they saw the sun touch the horizon, they stopped the watches and recorded the times. This measurement is all we need to calculate the Earth's radius.

Although the picture above isn't quite to scale with our experiment, r represents the radius of the Earth and h represents the height of the standing person with the stopwatch.  We know that in 24 hours, the earth makes a rotation of  360 degrees. With this information, we can set up a very simple relationship to find the change in angle of the sun we observed.  Let t be the measured time, and note that there are 86,400 seconds in 24 hours.
$\frac{t}{86400}=\frac{\theta}{2\pi}$
Once we have theta, we can use that to find r using the following equation:

$cos\theta=\frac{r}{r+h}$

Using the measurements t = 10 seconds and h = 1.9 meters, I found that r = 6807.225 (roughly 6,800) kilometers.  The actual radius of the earth is 6,378.1 kilometers. Our estimation was close (only about 7% off!).

There were a couple factors that contributed to the possible errors of this calculation. The first is human error. With more people trying to take the same measurements, we could have eliminated some of the error by averaging out our times. The atmosphere wasn't very clear on that day, which affected our view of the sun and made it more difficult to see when the sun touched the horizon.

Overall, I feel like the lab was a success. Not to mention that it was the most fun I've had doing a Caltech lab! I may just be biased as a Californian, but there's nothing that compares to the Pacific Ocean.

#### 1 comment:

1. Your picture of sunset over the ocean turned out gorgeous :)

Could you simplify the equation cos theta = r / (r+h) using the small angle approximation?

Good job!