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Fundamentals Of Physics
Found in: Page 10

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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 5.2×106 m

See the step by step solution

Step by Step Solution

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" d2+r2=r+h2 =r2+2rh+h2

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

d22rh … (i)

Step 3: Determination of the angle

Convert 24 into seconds.

1 hr = 3600 sec


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

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

θ360=11.186,400 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 tanθ

Squaring both the sides,

d2=r2tan2θ… (ii)

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

r=2htan2θ… (iii)

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

r=2×1.7 mtan20.04625=5.2×106 m

Thus, the radius of the earth is 5.2×106 m

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