Two of Jupiter’s moons have orbit radii which differ by a factor of 2. Their periods

Differ by a factor of $2 \sqrt{2}$

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Differ by a factor of $\sqrt{2}$

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Differ by a factor of 4

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Are the same

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Solution

The correct option is A
Differ by a factor of $2 \sqrt{2}$

Step 1: Given data and assumptions made:
Let, the radius of Jupiter’s first moon is R₁and the time period T₁ and the radius of Jupiter’s second moon is R₂ and the time period T₂.
$R_{1} = 2 R_{2}$
Step 2: Calculating time period
Kepler's Third Law: The square of the orbital period of the planet is directly proportional to the cube of the semi-major axis of its orbit.
By using Kepler’s third law, $T^{2} a R^{3}$ .
Therefore, the ratio of their time period will be,
$\frac{{T_{1}}^{2}}{{T_{2}}^{2}} = \frac{{R_{1}}^{3}}{{R_{2}}^{3}} One moon orbit radii is differ by another one by a factor of 2 . So, R_{1} = 2 R_{2} \frac{{T_{1}}^{2}}{{T_{2}}^{2}} = \frac{{(2 R_{2})}^{3}}{{R_{2}}^{3}} \frac{{T_{1}}^{2}}{{T_{2}}^{2}} = {(2)}^{3} \frac{T_{1}}{T_{2}} = (\sqrt{8}) \frac{T_{1}}{T_{2}} = 2 \sqrt{2} T_{1} = 2 \sqrt{2} T_{2}$
Hence, the correct answer is option (a).

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