Under some circumstances, a star can collapse into an...

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly $10^{14}$ times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was $9.0 \times 10^{5} \mathrm{~km}$ (comparable to our sun); its final radius is $16 \mathrm{~km}$. If the original star rotated once in 32 days, find the angular speed of the neutron star.

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Key Concepts

Moment of Inertia for a Uniform Solid Sphere

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation. For a uniform solid sphere, the moment of inertia is proportional to the mass times the square of its radius, which means that even a small change in the radius can lead to a significant change in the moment of inertia.

Angular Speed

Angular speed describes how fast an object rotates, typically measured in radians per second. It is directly related to the rotation period and is affected by any changes in the moment of inertia when angular momentum is conserved, such as in stellar collapse scenarios.

Conservation of Angular Momentum

This principle states that if no external torques act on a system, its total angular momentum remains constant despite changes in the system's configuration. For collapsing astronomical bodies, such as a star turning into a neutron star, the reduction in the moment of inertia leads to a compensatory increase in angular speed to maintain the angular momentum.

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