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Io is a moon of Jupiter with a semi-major axis of $421{,}700\text{ kilometers}$, a mass of $8.9319×{ 10 }^{ 22} \text{ kilograms}$, a period of $1.769\text{ days}$ and an eccentricity of $0.0041$.

The average speed of Io in its orbit is $62{,}400\text{ kilometers/hour}$ and the average radius of the orbit is $422{,}000\text{ kilometers}$.

The orbit of Io is an ellipse with Jupiter at one of the foci. The semi-major axis is measured either from Point A or Point B to the center in the drawing below. Point A is where Io is at its greatest distance from Jupiter measured on the semi-major axis and Point B is when Io is closest to Jupiter along the semi-major axis.

The shape of the ellipse is determined by the eccentricity, $e=\frac { c }{ a }$. Calculating with the semi-major axis tells us that $c=1{,}728{,}970\text{ m}$. This is the distance from the center to Jupiter.

The rotational inertia for an object in orbit around the planet is $I=m{ r }^{ 2 }$.

Which of the following indicates the best value for the angular velocity of Io when it is moving the fastest?

A

$\omega =1.49\times { 10 }^{ -5 }\text{ rad/s}$

B

$\omega =4.14\times { 10 }^{ -5 }\text{ rad/s}$

C

$\omega =2.42\times { 10 }^{ 4 }\text{ rad/s}$

D

$\omega =7.30\times { 10 }^{ 12 }\text{ rad/s}$

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