In the context of conservation of angular momentum, a fascinating example is the life cycle of a star, particularly when it undergoes gravitational collapse at the end of its life. As a star exhausts its stellar energy, it experiences a significant reduction in both mass and radius. This scenario provides a clear illustration of how angular momentum is conserved, despite the changes in the star's physical properties.

For instance, consider our sun, which has a rotation period of approximately 24.5 days. If the sun were to collapse, losing 90% of its mass and radius, we can analyze the implications for its new period of rotation. The new mass, denoted as \( m' \), would be 10% of the original mass, and the new radius would also be 10% of the original radius. To find the new period of rotation, we can apply the principle of conservation of angular momentum.

Angular momentum (\( L \)) can be expressed as:

\[ L = I \cdot \omega \]

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For a solid sphere, the moment of inertia is given by:

\[ I = \frac{1}{2} m r^2 \]

Since angular velocity (\( \omega \)) is related to the period (\( T \)) by the equation:

\[ \omega = \frac{2\pi}{T} \]

Substituting this into the angular momentum equation, we can express it in terms of the period:

\[ L = \frac{1}{2} m r^2 \cdot \frac{2\pi}{T} \]

When the star collapses, both mass and radius decrease, leading to a new angular momentum equation. The initial angular momentum can be set equal to the final angular momentum, allowing us to analyze the proportional changes. The mass and radius decrease by a factor of 0.1, and since the radius is squared in the moment of inertia formula, the effective change in angular momentum results in a factor of \( 0.1^3 = 0.001 \). This indicates that the angular momentum decreases by a factor of 1000.

Consequently, to maintain conservation of angular momentum, the period of rotation must increase by a factor of 1000. Thus, the new period of rotation can be calculated as:

\[ T' = \frac{1}{1000} T \]

Substituting the initial period of 24.5 days, we find:

\[ T' = \frac{24.5 \text{ days}}{1000} \approx 0.0245 \text{ days} \]

To convert this into minutes, we multiply by the number of hours in a day (24) and then by the number of minutes in an hour (60):

\[ T' \approx 0.0245 \times 24 \times 60 \approx 35 \text{ minutes} \]

This calculation reveals that after the sun collapses, it would complete a full rotation in approximately 35 minutes, illustrating the dramatic effects of gravitational collapse on a star's rotational dynamics.