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The Saturn is about six times farther fr...

The Saturn is about six times farther from the Sun than The Mars Which planet has
(a) the greater period of revolution
(b) the greater orbital speed and (c )the greater angular speed ?

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The correct Answer is:
To solve the problem, we need to analyze the relationship between the distance of the planets from the Sun and their orbital characteristics based on Kepler's laws of planetary motion. ### Given: - Saturn is about 6 times farther from the Sun than Mars. - Let the distance of Mars from the Sun be \( r_m \). - Therefore, the distance of Saturn from the Sun is \( r_s = 6r_m \). ### (a) Greater Period of Revolution: According to Kepler's Third Law, the square of the period of revolution \( T \) of a planet is directly proportional to the cube of the semi-major axis (orbital radius) of its orbit. Mathematically, this is expressed as: \[ T^2 \propto r^3 \] This implies: \[ T \propto r^{3/2} \] Since Saturn is farther from the Sun than Mars, we have: \[ r_s > r_m \] Thus, the period of revolution for Saturn will be greater than that of Mars: \[ T_s > T_m \] **Conclusion:** Saturn has the greater period of revolution. ### (b) Greater Orbital Speed: The orbital speed \( v \) of a planet is given by the formula: \[ v = \sqrt{\frac{GM}{r}} \] Where: - \( G \) is the gravitational constant, - \( M \) is the mass of the Sun, - \( r \) is the distance from the Sun. From this formula, we see that the orbital speed is inversely proportional to the square root of the radius: \[ v \propto \frac{1}{\sqrt{r}} \] Since \( r_s = 6r_m \): \[ v_s \propto \frac{1}{\sqrt{6r_m}} < v_m \propto \frac{1}{\sqrt{r_m}} \] Thus, the orbital speed of Mars is greater than that of Saturn: \[ v_m > v_s \] **Conclusion:** Mars has the greater orbital speed. ### (c) Greater Angular Speed: The angular speed \( \omega \) is related to the period of revolution by the formula: \[ \omega = \frac{2\pi}{T} \] This indicates that angular speed is inversely proportional to the period of revolution: \[ \omega \propto \frac{1}{T} \] Since we established that the period of revolution for Saturn is greater than that for Mars: \[ T_s > T_m \implies \omega_s < \omega_m \] Thus, Mars has a greater angular speed than Saturn: \[ \omega_m > \omega_s \] **Conclusion:** Mars has the greater angular speed. ### Final Answers: (a) Saturn has the greater period of revolution. (b) Mars has the greater orbital speed. (c) Mars has the greater angular speed.
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