Question

Mass of Saturn. The innermost rings of Saturn orbit in a circle with a radius of 67,000 kilometers at a speed of \(23.8 \mathrm{km} / \mathrm{s}\). Use the orbital velocity formula to compute the mass contained within the orbit of those rings. Compare your answer with the mass of Saturn listed in Appendix E.

Short answer

The mass within Saturn's rings is approximately \(5.64 \times 10^{26} \text{ kg}\), in close agreement with Saturn's mass.

Step-by-step solution

Understand the Orbital Velocity Formula

The orbital velocity formula is expressed as \( v = \sqrt{\frac{GM}{r}} \), where \( v \) is the orbital velocity, \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \), \( M \) is the mass of the celestial body, and \( r \) is the radius of the orbit.

Rearrange the Formula to Solve for Mass

To find the mass \( M \), rearrange the formula: \( M = \frac{v^2 r}{G} \). This formula allows us to solve for the mass using the given orbital velocity and radius.

Convert Radius to Meters

The radius is given in kilometers (67,000 km). Convert it to meters: \( 67,000 \, \text{km} = 67,000,000 \, \text{m} \).

Plug in Known Values

Substitute the known values into the equation: \[ M = \frac{(23.8 \, \text{m/s})^2 \times 67,000,000 \, \text{m}}{6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2} \].

Calculate the Mass

Perform the calculation: \[ M = \frac{566.44 \, \text{m}^2/ ext{s}^2 \times 67,000,000 \, \text{m}}{6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2} \]. Simplifying, \( M \approx 5.64 \times 10^{26} \text{ kg} \).

Compare with Saturn's Mass

The calculated mass of \( 5.64 \times 10^{26} \text{ kg} \) closely matches the actual mass of Saturn (\( 5.68 \times 10^{26} \text{ kg} \) as listed in various appendices).

Key concepts

mass of Saturn

Saturn, the sixth planet from the Sun, is known for its stunning rings and its massive size when compared to Earth. The actual mass of Saturn is pivotal in understanding its gravitational pull and its ability to hold the intricate ring system with remarkable ease. Saturn's mass is approximately \(5.68 \times 10^{26} \text{ kg} \). This immense mass is a key factor allowing it to have multiple moons and dominate the gravitational landscape of its surroundings.
Besides its role in celestial mechanics, understanding the mass of Saturn helps us deduce various characteristics of its rings. The exercises like the one originally provided challenge us to use orbital parameters to calculate this mass through orbital motion, illustrating the balance in physics governing these gigantic celestial objects.

gravitational constant

To understand the gravitational forces at play in Saturn’s orbit, we need to consider the gravitational constant \( G \), a fundamental pillar in celestial mechanics. This constant is approximately \(6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\). It serves as a key component in Newton's law of universal gravitation, which states that every mass attracts every other mass in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• In Calculations: The gravitational constant remains a critical factor in calculations involving celestial bodies.
• Universal Relevance: It is consistent throughout the universe, allowing scientists to make precise calculations.

Without \( G \), calculating the mass of celestial bodies, like Saturn, from observed motions would be impossible, emphasizing its importance in physics.

celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions and orbits of celestial objects, which are influenced by gravitational forces. This field encompasses the analysis of planetary orbits, trajectories of comets, and even the movement of man-made satellites.

• Orbital Velocity: This is a crucial concept in celestial mechanics. For Saturn's rings, the orbital velocity tells us how fast the rings must move to remain in stable orbit.
• Kepler’s Laws of Planetary Motion: These laws describe how celestial objects move in orbits due to gravitational attraction, a foundation of celestial mechanics.

Studying celestial mechanics allows us to predict the movement and position of planetary bodies, essential for navigation in space exploration and understanding the dynamics of our solar system.

physics formula derivation

When solving problems in physics, deriving the correct formulas is vital. For instance, finding the mass of Saturn from the orbital motion of its rings involves rearranging the orbital velocity formula to suit our needs. Original formulas, like \( v = \sqrt{\frac{GM}{r}} \), give us a starting point with known quantities like velocity (\( v \)), and radius (\( r \)). However, to determine the mass \( M \), we rearrange it to \( M = \frac{v^2 r}{G} \).

• Rearrangement: This process involves placing the known values in a position to isolate the desired unknown variable.
• Understanding Variables: Knowing what each symbol in a formula represents is crucial, ensuring correct application and interpretation.

Effective derivation of formulas and equations is essential for solving physics problems accurately and is a critical skill to master for any student delving into scientific studies.