Keplers Laws

Answer (e) Angular momentum

Given, Semi-major axis of the orbit of saturn Where, semi major axis of earth According to Kepler's law, Let the time period of revolution of saturn around the sun is year years

Concept: Kepler’s third law:

Step 1: Apply relation.


Step 2: Solve.

Step 1: Understanding the Concept:
A satellite spinning and revolving has moment of inertia, potential energy, rotational KE, and angular momentum.

Step 2: Detailed Explanation:
1. Moment of inertia — exists due to mass distribution.
2. Potential energy — due to gravitational field.
3. Rotational kinetic energy — due to spinning.
4. Vibrational energy — no vibration mentioned, so absent.
5. Angular momentum — due to both spin and orbit.

Step 3: Final Answer:
The satellite does not have vibrational energy.

Step 1: Understanding the Concept:
Kepler's laws describe planetary motion.

Step 2: Detailed Explanation:

• Kepler's First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus.
• Kepler's Second Law (Law of Equal Areas): The line joining a planet to the Sun sweeps out equal areas in equal times.
• Kepler's Third Law (Law of Periods): The square of the orbital period is proportional to the cube of the semi-major axis.

Step 3: Final Answer:
This statement is Kepler's second law.

Step 1: Understanding the Concept:
The problem relates the time period of revolution of a planet to its orbital radius (semi-major axis). This relationship is governed by Kepler's Third Law of Planetary Motion.

Step 2: Key Formula or Approach:
Kepler's Third Law states that the square of the time period ( ) of a planet is directly proportional to the cube of the semi-major axis ( ) of its orbit:

For two planets, say Earth ( ) and Saturn ( ), we can set up a ratio:


Step 3: Detailed Explanation:
Let's denote the parameters for Earth with subscript 'E' and for Saturn with subscript 'S'.
We know the time period of Earth, year.
The problem states that the semi-major axis of Saturn is nine times that of Earth:

Now, substitute these values into Kepler's Third Law equation:


To find , we take the square root of both sides:

We can simplify this by rewriting as :


So, the time period of Saturn is approximately 27 Earth years.

Step 4: Final Answer:
The time period of revolution of Saturn is 27 years.

Concept:
• Angular momentum conservation:

Step 1: Given ratio


Step 2: Apply conservation
Let nearest speed , so:

Step 3: Solve
Final Conclusion:
Option (C)

Step 1: Understanding the Concept:
The relationship between the orbital time period of a satellite and its orbital radius is governed by Kepler's Third Law of Planetary Motion.

Step 2: Key Formula or Approach:
Kepler's Third Law states that the square of the time period ( ) is directly proportional to the cube of the orbital radius ( ):

For two different orbits around the same central body:


Step 3: Detailed Explanation:
Let the initial orbit be: radius , time period .
Let the new orbit be: radius , time period .
Using the ratio formula:



Multiply both sides by :

Take the square root of both sides:

Simplify the radical ( ):


Step 4: Final Answer:
The new time period is .

Step 1: Understanding the Concept:
According to Kepler's Second Law (Law of Areas), a line joining a planet and the sun sweeps out equal areas during equal intervals of time. A direct consequence of this is the conservation of angular momentum for the planet as it orbits the sun.

Step 2: Key Formula or Approach:
The angular momentum of a planet of mass moving with velocity at a distance from the sun is given by .
At the maximum distance (aphelion) and minimum distance (perihelion), the velocity vector is strictly perpendicular to the position vector, so .
Therefore, conservation of angular momentum simplifies to:


Step 3: Detailed Explanation:
The problem text contains an obvious typographical error: "minimum velocity is and minimum velocity is ".
Based on the physical laws, velocity is inversely proportional to distance. Therefore:
- At maximum distance , the planet moves slowest. Let this minimum velocity be .
- At minimum distance , the planet moves fastest. Let this maximum velocity be .
Applying the conservation of angular momentum:

Cancel the mass :

We need to find the ratio , which is :


Step 4: Final Answer:
The ratio is .