Question

Kepler's second law regarding constancy of aerial velocity of a palnet is consequence of the law of conservation of (A) energy (B) angular Momentum (C) linear momentum (D) None of these

Short answer

Kepler's second law regarding constancy of aerial velocity of a planet is a consequence of the law of conservation of (B) Angular Momentum. This conservation law is relevant to Kepler's second law because the gravitational force from the star does not change the planet's angular momentum, thus maintaining the constant aerial velocity.

Step-by-step solution

Understanding Kepler's second law

Kepler's second law states that a line connecting a planet to its star sweeps out equal areas in equal time intervals. In other words, the aerial velocity of the planet, the rate at which it sweeps out area, remains constant.

Reviewing conservation laws

Let's briefly review the conservation laws mentioned in the options: (A) Conservation of energy: The total energy of an isolated system remains constant unless acted upon by an external force. In the case of a planet orbiting a star, gravitational potential energy and kinetic energy are constantly changing, but their sum, the total mechanical energy, remains constant. (B) Conservation of angular momentum: The total angular momentum of an isolated system remains constant unless acted upon by an external torque. Angular momentum is given by the product of an object's moment of inertia and its angular velocity. (C) Conservation of linear momentum: The total linear momentum of an isolated system remains constant unless acted upon by an external force. Linear momentum is given by the product of an object's mass and its velocity. Now, let's determine which of these conservation laws is related to Kepler's second law.

Connecting Kepler's second law with conservation of angular momentum

Kepler's second law deals with the constant aerial velocity of a planet orbiting a star. Since we're considering changes in a planet's position with respect to the star, the relevant conservation law must deal with rotational motion. Among the available options, conservation of angular momentum is the most appropriate choice. When a planet moves around its orbit, the force acting on it is the gravitational force from the star, which is always toward the center of the circular path. This force causes a centripetal acceleration but no change in the planet's angular momentum since there is no external torque acting on the system. As a result, the planet's aerial velocity remains constant, providing a basis for Kepler's second law. Therefore, the answer is: (B) Angular Momentum

Key concepts

Conservation of Angular Momentum

Imagine spinning a toy top. You have to give it a spin, and once you do, it keeps spinning until something stops it. This continuous spinning is due to a property called angular momentum. Angular momentum is a measure of how much motion is contained in an object's rotation. For objects like planets that travel in curved or circular paths, angular momentum is conserved. This means, unless there is an outside force, such as a push or a pull, the amount of angular momentum stays the same throughout motion.

For Kepler's Second Law, which tells us that planets sweep out equal areas in equal times as they orbit the Sun, we see this property in action. The gravitational pull from the sun causes a planet to move in a specific path, but crucially, it directly affects only the direction of the planet's velocity, not its angular momentum. Because there is no additional force acting to spin or slow the planet, the planet's angular momentum is conserved, maintaining its constant aerial speed as it moves along its path.

Aerial Velocity

Aerial velocity sounds complex, but it's quite manageable once you break it down. Picture a spinning figure skater pulling in her arms; she spins faster, but she covers less ground. In planetary terms, aerial velocity is about how fast a planet covers an area as it orbits the Sun.

Kepler's Second Law states that this aerial velocity is constant, meaning a planet moves in such a way that it always covers equal areas in equal time periods. When a planet is closer to the Sun, it speeds up, sweeping out a short wide arc. When it's farther, it slows down, covering a long narrow arc. Despite these speed changes, the area covered in the given timeframe remains the same. This simple, yet profound, concept reinforces how interconnected speed and position are in orbital motion.

Planetary Motion

Planetary motion is the graceful dance that planets perform as they travel through the solar system. Governed by the laws of motion and the influence of gravity, planetary paths are elliptical, meaning each planet follows an oval-shaped orbit around the Sun.

This kind of movement is not random; it's the result of gravitational forces and the initial conditions of the planet's birth. Influenced by these forces, a planet's velocity constantly changes in magnitude and direction. The Conservation of Angular Momentum plays a vital role here, ensuring that despite these changes, the overall motion remains predictable and balanced.

• Elliptical Orbits: Planets don't orbit the Sun in perfect circles; instead, they follow elliptical paths.
• Gravitational Pull: The Sun's gravity is the central force that guides this motion.
• Velocity Change: A planet's speed changes; it moves faster when closer to the Sun and slower when further away due to gravitational forces.

Understanding planetary motion isn't just about memorizing facts; it's about seeing the complexity and harmony within our solar system.