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Chapter 6: Problem 784

The time period of a satellite of earth is 5 hours. If the separation between the earth and the satellite is increased to four times the previous value, the new time period will become \(\ldots \ldots \ldots\) hours (A) 10 (B) 120 (C) 40 (D) 80

Short Answer

Expert verified
The new time period of the satellite will be 40 hours when the separation between the Earth and the satellite is increased to four times the previous value.

Step by step solution

01

Write the proportionality equation

Since the square of the period is proportional to the cube of the semi-major axis, we can write the equation as: \(T_1^2 / T_2^2 = a_1^3 / a_2^3\) where - \(T_1\) is the initial time period, - \(T_2\) is the new time period, - \(a_1\) is the initial separation between the Earth and the satellite, and - \(a_2\) is the new separation (four times the initial value).
02

Plug in values and solve for the new time period

We are given the initial time period \(T_1 = 5 \) hours and the new separation \(a_2 = 4a_1\). We need to find \(T_2\). Now we can set up the proportionality equation: \(\frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3}\) Substitute the given values: \(\frac{(5)^2}{T_2^2} = \frac{a_1^3}{(4a_1)^3}\) Simplify the equation: \(\frac{25}{T_2^2} = \frac{a_1^3}{64a_1^3}\) Now cross multiply: \(25 \cdot 64a_1^3 = T_2^2 \cdot a_1^3\) Divide by \(a_1^3\) and simplify: \(25 \cdot 64 = T_2^2\) Calculate the square root of both sides: \(T_2 = \sqrt{25 \cdot 64}\) \(T_2 = 5 \cdot 8 = 40\) So the new time period will be 40 hours. The correct answer is (C) 40.

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