11.11
Conservación del Momentum Angular: Aplicación
El momentum angular total de un sistema permanece constante si el esfuerzo de torsión externo neto que actúa sobre el sistema es cero. Ejemplos de tal…
Para cualquier objeto que gire alrededor de un eje de rotación, la conservación del momento angular se mantiene si no actúa sobre él ningún par externo.
Por ejemplo, supongamos que el Sol, que tiene una velocidad angular de dos coma seis por diez elevado a menos seis radianes por segundo, colapsa en una enana blanca tal que su radio disminuye en un factor de quinientos. Suponiendo que la masa perdida no se lleve ningún momento angular, ¿cuál será la energía cinética rotacional final de la enana blanca?
Aquí las cantidades conocidas son los radios inicial y final, las masas inicial y final, y la velocidad angular del Sol. La cantidad desconocida es la energía cinética rotacional final de la enana blanca.
Aquí, la conservación del momento angular se mantiene, y asumiendo que el Sol y la enana blanca tienen densidades esféricas uniformes, sustituyendo su momento de inercia, se puede calcular la velocidad angular final de la enana blanca.
La energía cinética rotacional de la enana blanca se puede calcular sustituyendo el valor de la velocidad angular final.
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Q1: When does conservation of angular momentum apply to a rotating system?
Conservation of angular momentum applies when no external torque acts on a rotating system. A system's total angular momentum remains constant if the net external torque is zero. Examples include freely spinning objects in space or systems where friction and other external forces are negligible, allowing the angular momentum to be preserved.
Q2: How does angular velocity change when a rotating object's moment of inertia decreases?
When a rotating system's moment of inertia decreases, angular velocity must increase to conserve angular momentum. This relationship follows from the conservation principle: if the radius of rotation decreases, the angular velocity increases proportionally. Tornadoes exemplify this—as rotating storm systems contract, their angular velocity increases dramatically.
Q3: What happens to a collapsing star's rotation rate according to angular momentum conservation?
When a star collapses, its radius decreases significantly while its mass remains essentially constant. As the moment of inertia decreases, the star's angular velocity increases substantially to conserve angular momentum. For example, if the Sun collapsed into a white dwarf with radius reduced by a factor of 500, its rotation rate would increase dramatically.
Q4: How can you calculate rotational kinetic energy after angular momentum is conserved?
After determining the final angular velocity using conservation of angular momentum, substitute this value into the rotational kinetic energy formula. For a spherical object with uniform density, calculate the moment of inertia, then apply the kinetic energy equation. This approach connects angular momentum conservation to energy calculations using the work energy theorem for rotational motion.
Q5: Why do astronauts in space maintain zero angular momentum while twisting their bodies?
Astronauts floating inside a spacecraft experience zero external torque when they don't push against the vessel walls. Without external torque, their angular momentum remains conserved at zero. They can twist and reorient their bodies through internal motions, but their total angular momentum relative to the spacecraft stays zero.
Q6: How does the solar system's formation demonstrate angular momentum conservation?
The solar system formed from a large rotating cloud of gas and dust. Gravitational forces caused the cloud to contract, decreasing its radius. As the cloud contracted, its angular velocity increased due to conservation of angular momentum, eventually forming the rotating solar system we observe today.
Q7: What role does moment of inertia play in angular momentum conservation problems?
Moment of inertia determines how angular velocity changes when angular momentum is conserved. For uniform spherical objects, moment of inertia depends on mass and radius. When solving conservation problems, calculate initial and final moments of inertia, then use the conservation equation to find the final angular velocity and subsequent rotational kinetic energy.