Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed o

step 1 So this is a star. It's collapsing. And basically it's a conservation of angular momentum proble

Suppose a star the size of our Sun, but with mass 8.0 times...

Key Concepts

Gravitational Collapse

Gravitational collapse refers to the process in which an astronomical object undergoes a significant reduction in size as it contracts under its own gravity. In the context of stellar evolution, gravitational collapse plays a fundamental role in the formation of compact objects such as neutron stars and black holes. During collapse, the conservation laws of physics, especially those of energy and angular momentum, are critical in determining the final state and properties of the collapsed object.

Conservation of Angular Momentum

The conservation of angular momentum is a pivotal principle in understanding how rotation speeds change when the dimensions of a system are altered. When an object contracts — as in the collapse to a neutron star — its moment of inertia decreases dramatically. According to the conservation of angular momentum, if no external torque acts on the system, the product of the moment of inertia and the rotational speed must remain constant, leading to a significant increase in the rotational speed as the object shrinks.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on both its mass distribution and geometry. For a uniform sphere, the moment of inertia is calculated using a specific formula. In stellar collapse scenarios, the change in moment of inertia as the radius decreases is central to determining how the rotational dynamics of the star will adjust, making it a key factor in calculating the final rotation speed.

Uniform Sphere Assumption

The uniform sphere assumption simplifies the complex reality of a star's internal structure by considering its mass as evenly distributed throughout its volume. This assumption is useful for theoretical calculations, as it allows for the application of standard formulas for moment of inertia and other physical properties. Although real stars may have non-uniform densities, using a uniform sphere model provides a tractable way to analyze the effects of collapse on rotation.

Effect of Mass Loss on Angular Momentum

In scenarios involving gravitational collapse, it is critical to consider how mass loss impacts the total angular momentum of the system. When a significant portion of a star's mass is expelled, the carrying away of angular momentum can alter the dynamics of the remaining core. Depending on whether the lost mass carries no angular momentum or retains a proportional share relative to its mass, the final angular momentum—and thus the rotational speed—of the collapsed object can vary significantly.

Transcript

00:01 So this is a star.

00:03 It's collapsing.

00:06 And basically it's a conservation of angular momentum problem where they're trying to, where they're asking us for the final angular velocity, such that the thrown -off mass has no angular momentum, and the thrown -off mass has three -fourth of the initial angular momentum.

00:23 So we should first just write down our givens.

00:29 So we'll say the mass of the star, well, denoted as s equals eight times the mass of the sun.

00:38 And the radius of the star equals the radius of the sun.

00:42 We know that the angular velocity of the star for its revolution is one revolution per every nine days.

00:57 The mass of the star final equals the mass of the star over four, or simply two times.

01:05 The mass of the sun.

01:09 The radius of the star final is going to equal 12 kilometers or we should convert to 12 ,000 meters.

01:20 Now one essentially one thing that we should find in this case would be well first of all we should apply the the for part a apply the conservation of momentum so thrown off mass has no l, angular momentum.

01:50 So, l .i equals l final, and that means that i initial, omega initial equals i final, omega, final.

02:01 So that means that omega final equals omega initial, i initial over i final.

02:09 Now i initial is going to be equal to two times the mass of the s times the radius of s squared all over 5.

02:23 Or we can say 16 times the mass of the sun, radius of the sun squared divided by 5.

02:36 I final is going to be equal to two times the mass of the sun of the star times r s the radius of the star final final over five or four mass of the sun the radius of the sun squared or sorry this is not the radius of the sun squared anymore this would actually become 12 ,000 squared divided by five and i initial over i final if you were to simply divide these you would actually get to four radius of the sun squared divided by 12 ,000 squared again these five cancel out the mass of the sun cancels out 16 divided by four equals four that's how that's accounted for there the radius of the sun is here and then 12 ,000 would be that be left as a denominator so at that point we can just say that omega, we'll just separate this a bit, omega final equals one revolution in nine days times four times the radius of the sun 6 .957 times 10 to the eighth squared divided by 12 ,000 squared...

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