Final answer:
The conservation of angular momentum is a principle stating that the total angular momentum of a system remains constant if no external torque is applied. Kepler's second law is a manifestation of this conservation, describing how a planet's velocity changes along its orbit to maintain constant angular momentum. This principle underpins various phenomena from figure skating to the formation of solar systems.
Step-by-step explanation:
Conservation of Angular Momentum and Kepler's Second Law
The conservation of angular momentum is a principle in physics that states when no net external torque acts on a system, the total angular momentum of the system remains constant. In mathematical terms, angular momentum L is given by L = Iω, where I is the moment of inertia and ω is the angular velocity. For an isolated system, angular momentum can change only if an external torque is applied (net τ = ΔL).
Kepler's second law, which states that a planet sweeps out equal areas in equal times as it orbits the Sun, is a direct consequence of the conservation of angular momentum. When a planet is nearer to the Sun in its elliptical orbit, it must move faster to conserve angular momentum since its distance from the Sun (and thus its moment of inertia) is smaller. Conversely, when the planet is farther from the Sun, it moves more slowly. This areal constant speed ensures the conservation of angular momentum.
Everyday examples of this principle include figure skaters pulling their arms in to spin faster and extending them to slow down, which demonstrates how manipulating the moment of inertia affects rotational speed to conserve angular momentum. A cosmic scale example is the formation of our solar system from a rotating cloud of gas and dust, which contracted and spun faster due to this principle.