11.11
Conservation of Angular Momentum: Application
A system's total angular momentum remains constant if the net external torque acting on the system is zero. Examples of such systems include a freely…
For any object rotating about a rotational axis, the conservation of angular momentum holds if no external torque acts on it.
For example, suppose the Sun, having an angular velocity of two point six times ten to the power of negative six radians-per-second, collapses into a white dwarf such that its radius decreases by a factor of five hundred. Assuming that the lost mass carries away no angular momentum, what will be the white dwarf's final rotational kinetic energy?
Here the known quantities are initial and final radii, initial and final masses, and the angular velocity of the Sun. The unknown quantity is the final rotational kinetic energy of the white dwarf.
Here, the conservation of angular momentum holds, and assuming that the Sun and the white dwarf each have uniform spherical densities, substituting for their moment of inertia, the final angular velocity of the white dwarf can be calculated.
The rotational kinetic energy of the white dwarf can be calculated by substituting the value of the final angular velocity.
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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.