Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

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 ${5.2×10}^{6} m$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

The height of the person h = 1.70 m

Time elapsed t = 11.1 sec ,

Step 2: Understanding the concept of time with the rotation of the earth

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 $θ$ using these two different lines of sight. Using this value of $θ$ 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" $d^{2} + r^{2} = {(r + h)}^{2} = r^{2} + 2 r h + h^{2}$

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

$d^{2} \approx 2 r h$ … (i)

Step 3: Determination of the angle

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,

$\frac{θ}{360} = \frac{11.1}{86, 400} θ = 0.04625 degrees$

Thus, the angle $θ$ is $0.04625 degrees$

Step 4: Determination of the radius of the earth

From the figure, it can be written that,

$d = r \tan (θ)$

Squaring both the sides,

$d^{2} = r^{2} \tan^{2} (θ)$ … (ii)

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

$r = \frac{2 h}{\tan^{2} (θ)}$ … (iii)

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

$r = \frac{2 \times 1.7 m}{\tan^{2} (0.04625)} = 5.2 \times 10^{6} m$

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

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Given data

Step 2: Understanding the concept of time with the rotation of the earth

Step 3: Determination of the angle

Step 4: Determination of the radius of the earth

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Fundamentals Of Physics

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