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Q19P

Expert-verifiedFound in: Page 10

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**Suppose that, while lying on a beach near the equator watching the Sunset over a calm ocean, you start a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height H = 1.70 m, and stop the watch when the top of the Sun again disappears. If the elapsed time is t = 11.1 s, what is the radius r of Earth?**

The radius of the earth is ${\text{5.2\xd710}}^{6}m$

The height of the person h = 1.70 m

Time elapsed t = 11.1 sec ,

Considering the projection of the earth is nothing more thana circle. A person standing in the center of the circle can be seen as a line h. Drawing the tangent to the circle from the line h it forms a rectangular triangle. **Therefore, using the ****Pythagorean Theorem****, the radius of the earth can be calculated.**

When the person stands up, his line-of-sight changes. Find the angle $\theta $ using these two different lines of sight. Using this value of $\theta $ and the time in which this change took place, it is possible to find the radius of the earth.

From Pythagorean Theorem,

role="math" localid="1651835120648" ${d}^{2}+{r}^{2}={\left(r+h\right)}^{2}\phantom{\rule{0ex}{0ex}}={r}^{2}+2rh+{h}^{2}$

Here,*h* is very small as compared to *r*, so neglecting higher terms of *h*,

${d}^{2}\approx 2rh$ … (i)

Convert 24 into seconds.

1 hr = 3600 sec

Therefore,

Earth takes 24 hrs for one complete rotation, that is, 360 degree

So, for t = 11.1 s it takes *θ* degrees,

$\frac{\theta}{360}=\frac{11.1}{86,400}\phantom{\rule{0ex}{0ex}}\theta =0.04625\text{degrees}\phantom{\rule{0ex}{0ex}}$

Thus, the angle $\theta $ is $\text{0.04625 degrees}$

From the figure, it can be written that,

$d=r\mathrm{tan}\left(\theta \right)$

Squaring both the sides,

${d}^{2}={r}^{2}{\mathrm{tan}}^{2}\left(\theta \right)$… (ii)

Now substitute the value of from equation (i) to find the equation for .

$r=\frac{2h}{{\mathrm{tan}}^{2}\left(\theta \right)}$… (iii)

Substitute the values in equation (iii) to calculate the radius of the earth.

$r=\frac{2\times 1.7\text{m}}{{\mathrm{tan}}^{2}\left(0.04625\right)}\phantom{\rule{0ex}{0ex}}=5.2\times {10}^{6}\text{m}$

Thus, the radius of the earth is $5.2\times {10}^{6}\text{m}$

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