An object collapses to a black hole when escape speed at its...

An object collapses to a black hole when escape speed at its surface becomes the speed of light. At what radius does that occur for an object with (a) the Sun's mass and (b) the mass of a galaxy, approximately $10^{11}$ solar masses. (You can use Newtonian gravitation for this calculation.)

Key Concepts

Newtonian Gravitation

Newtonian gravitation describes the attraction between masses using the inverse square law. It provides the basis for calculating gravitational forces and potentials, which in turn are used to derive important quantities such as the escape velocity from a body’s surface.

Escape Velocity

Escape velocity is the minimum speed required for an object to break free from the gravitational pull of a mass without further propulsion. It is derived from equating the kinetic energy needed to overcome gravitational potential energy and is expressed as ?(2GM/R), where G is the gravitational constant, M is the mass, and R is the distance from the center of mass.

Schwarzschild Radius

The Schwarzschild radius is the critical radius at which the escape velocity equals the speed of light. In a simplified Newtonian picture, if an object's physical radius contracts to this value, not even light can escape its gravitational pull, effectively forming a black hole. This concept, while rooted in general relativity, provides a useful heuristic obtained from Newtonian calculations.

Mass Scaling in Gravitational Collapse

The condition for black hole formation based on escape velocity implies that the critical radius scales linearly with mass. More massive objects require a proportionally larger radius to reach the escape speed equal to the speed of light, underscoring that the black hole radius increases with mass. This scaling is key when comparing objects like a star and a galaxy in the context of gravitational collapse.

Transcript

00:03 Let's start from the party of this problem.

00:05 In this case, we have to find the radius of the object.

00:07 They turn into the black hole and whose mass is equal to the mass of the sun.

00:14 So we need to calculate r, which is the radius of the black hole.

00:22 In this case, we can apply the law of consideration of energy as 1 divided by 2m.

00:29 We escape square is equal to capital g, small m, capital m, capital m.

00:36 Divide by r here this m is the mass of the object which is going to escape from the black hole we escape is the velocity of the object g is gravitational constant m is the mass of object again which is this one capital m is the mass of the black hole and this r is the radius of that black call so from here we can write the equation for the black hole as r is equals to 2 capital gm divided by we escape square we call it equation number one let's set the values into this equation number one so here we have capital r is equals to 2 into 6 .67 multiply by turnous power minus 11 newton meter square per kg square into the mass of the black hole which is equal to the mass of the sun that is 1 .99 multiply return as power 30 kg divided by waist cap which is equal to the speed of light so we have tried 3 .0 multiplied by turner as power 8 meter per second whole square from here we will get the value for this r as r is equals to 2 .95 kilometer so this is the radius of that block call now let's move to the part b of this problem in this case, we have to find the radius of the black hole if the whole galaxy is turned into the black hole.

02:28 So in this case, we have to find our prime...

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