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Chapter 6: Problem 800

According to keplar, the period of revolution of a planet ( \(\mathrm{T}\) ) and its mean distance from the sun (r) are related by the equation (A) \(\mathrm{T}^{3} \mathrm{r}^{3}=\) constant (B) \(\mathrm{T}^{2} \mathrm{r}^{-3}=\) constant (C) \(\mathrm{Tr}^{3}=\) constant (D) \(\mathrm{T}^{2} \mathrm{r}=\) constant

Short Answer

Expert verified
Based on the comparison, option (B) \(T^2r^{-3} = \) constant is the correct representation of Kepler's Third Law among the given options.

Step by step solution

01

Kepler's Third Law formula

Kepler's third law of planetary motion states that the ratio of a planet's orbital period squared to its mean distance from the sun cubed is constant for all planets. Mathematically, this can be represented as: \[\frac{T^2}{r^3} = \textit{constant}\] Step 1: Cross-check the options with the formula
02

Comparing Kepler's Third Law to the given options

Now, let us examine each of the options provided and find the correct equation: (A) \(T^3r^3\) = constant: This option inverts the power of both T and r and does not match Kepler's Third Law. (B) \(T^2r^{-3}\) = constant: This option is merely rewriting the formula for Kepler's Third Law by incorporating the negative exponent in r rather than having r in the denominator. (C) \(Tr^3\) = constant: This option incorrectly multiplies T by r cubed, which does not resemble the correct formula. (D) \(T^2r\) = constant: This option does not have the correct power for r, so it does not match Kepler's Third Law. Comparing the options to the correct formula, we can now determine the correct option:
03

Conclusion

Based on the comparison, option (B) \(T^2r^{-3} = \) constant is the correct representation of Kepler's Third Law among the given options.

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