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Chapter 6: Problem 796

If the earth is at one- fourth of its present distance from the sun the duration of year will be (A) half the present Year (B) one-eight the present year (C) one-fourth the present year (D) one-sixth the present year

Short Answer

Expert verified
The duration of a year, when the Earth is at one-fourth of its present distance from the sun, is one-eighth the present year. This corresponds to option (B).

Step by step solution

01

Consider Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet (in this case, Earth) is directly proportional to the cube of the semi-major axis (average distance from the sun) of its orbit. Mathematically, this can be expressed as: \(T^2 \propto a^3\)
02

Find the ratio of orbital periods

Let's denote the present orbital period of the Earth as \(T_1\) and its present distance from the sun as \(a_1\). Similarly, let the orbital period when the Earth is at one-fourth of its present distance be \(T_2\) and the distance from sun be \(a_2\). Since the Earth is at one-fourth of its present distance, we have \(a_2 = \frac{1}{4}a_1\). Now we need to find the ratio \( \frac{T_2}{T_1}\)
03

Apply Kepler's Third Law to find the ratio

We can write the ratio of the squared orbital periods as: \(\frac{T_2^2}{T_1^2} = \frac{a_2^3}{a_1^3}\) Since \(a_2 = \frac{1}{4}a_1\), we can substitute this value, and we get: \(\frac{T_2^2}{T_1^2} = \frac{(\frac{1}{4}a_1)^3}{a_1^3}\)
04

Simplify the equation and solve for the ratio

Simplify the above equation as: \(\frac{T_2^2}{T_1^2} = \frac{1}{64}\) Now we can take the square root of both sides to find the ratio of the orbital periods: \(\frac{T_2}{T_1} = \frac{1}{8}\)
05

Compare the ratio with the given options

The ratio \(\frac{T_2}{T_1} = \frac{1}{8}\) indicates that the duration of a year, when the Earth is at one-fourth of its present distance from the sun, is one-eighth of the present year. This corresponds to option (B): (B) one-eighth the present year.

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