September 2015 saw the historic discovery of gravitational waves, almost exactly 100 years after Einstein predicted their existence as a consequence of his theory of general relativity. Gravitational waves are a literal stretching and compressing of the fabric of space. Even the most sensitive instruments—capable of sensing that the path of a 4-km-long laser beam has lengthened by one-thousandth the diameter of a proton—can detect waves created by only the most extreme cosmic events. The first detection was due to the collision of two black holes more than 750 million light years from earth. Although a full description of gravitational waves requires knowledge of Einstein's general relativity, a surprising amount can be understood with the physics you've already learned. (d) Two black holes collide and merge when their Schwarzchild radii overlap; that is, they merge when their separation, which we've defined as 2r, equals 2RSch . Find an expression for ΔE=Ef−Ei , where Ei ≈ 0 because initially the black holes are far apart and Ef is their total energy at the instant they merge. This is the energy radiated away as gravitational waves. Your answer will be a fraction of Mc², and you probably recognize that this is related to Einstein's famous E=mc² . The quantity Mc² is the amount of energy that would be released if an entire star of mass M were suddenly converted entirely to energy.

Question

Accepted Answer

Hello, fellow physicists today, we're going to solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Scientists expect the release of gravitational waste and energy when two black holes merge, the merger of two black holes is conceivable when they are so close to one another in such a way that they cannot escape their gravitational attraction. A sophomore is asked to use the concepts of classical mechanics to find the energy variation delta E of two black holes. A mass capital M subscript BMB each during their merger, the energy variation is defined as the difference between the energy at the instant the merge occurs and the energy at the instant black holes were very far from each other. What does the sophomore report in terms of M BC squared M being multiplied by C squared hint the merger occurs when the separation distance between the black holes is twice the Schwitz allowed radius and the black holes can be treated as point like objects. OK. So we're trying to find our end goal is to report in terms of MB multiplied by C squared. So let's read off our multiple choice sensors to see what our final answer might be. A is negative MB multiplied by C squared B is negative MB multiplied by C squared divided by three C is negative MB multiplied by C squared divided by four and D is negative MB multiplied by C squared divided by eight. OK. So first off, let us recall and use the equation for the energy variation for the initial and final energy conditions which states that delta E is equal to the final energy minus the initial energy. Awesome. So the change in energy states that the final energy minus the initial energy. Awesome. So let us begin by determining the energy capital E when the two black holes are separated by a distance of two multiplied by capital R. So let us also recall that all celestial objects such as planets and black holes rotate due to this fact. Let us assume that at the instant the merger takes place that the two black holes are rotating about an axis passing through its center of mass. Note that the two black holes have the same mass MB, their center mass lies with midway between the two black holes. We can recall and express their energy capital E energy capital, E in terms of the kinetic and potential energy as let's call it equation one as E the energy is equal to the kinetic energy plus the potential energy between the two black holes. OK. We also need to recall the gravitational potential energy is, and let's call that equation two is U is equal to negative the gravitational constant multiplied by MB multiplied by MB divided by two R. OK. And let's remember that K is the kinetic energy about the center of the mass. OK. So note that each black hole has the following rotational kinetic energy. So let's make a note here. So the rotational kinetic energy is one half multiplied by capital I multiplied by Omega squared. So the final angular frequency or the displacement. So the final angle, angular frequency omega subscript F squared. OK. So, and we also need to note that the total energy of the system for the two black holes is, and let's this is an important one. So it's called equation three is the kinetic energy is equal to two multiplied by one half multiplied by I multiplied by the angular frequency or sometimes it's called the displacement square. And I, in this case, since we haven't mentioned yet, I is the moment of inertia about the center of mass. OK. So note that the equation for a black hole should be treated like a point like object. So treating it like a point like point like objects object, we can state that the moment of inertia is equal to MB multiplied by capital R squared where capital R is the distance separating each black hole from its center of mass. So at this stage, we need to find the angular frequency omega to do that. We need to recall and use Newton's second law in the radial direction. So let's call it equation five. And that states that the sum of capital F subscript R is equal to M multiplied by a subscript, lowercase R is equal to MB multiplied by V squared divided by capital R which is equal to MB multiplied by Omega multiplied by capital R where capital F subscript R is the gravitational force. And the formula for the gravitational force is fr is equal to the gravitational constant multiplied by MB squared divided by two multiplied by capital R all squared fantastic. So we can combine equations five and six to help us find the equation to determine what Omega will be. So we're trying to find the angular frequency. So note that MB multiplied by omega squared multiplied by R equal to the gravitational constant multiplied by M B squared divided by two capital R squared can be simplified too. So this is when we combine and set equation five equal to equation six, we can simplify it down to solve for just omega sol for the angular frequency. And it will state that the angular frequency squared is equal to the gravitational constant multiplied by MB divided by four RQ. OK. So now we can combine equations 34 and seven to get, let's call this one equation seven to get equation eight. So when we combine equations 34 and seven, we can get equation eight, which states that the kinetic energy is equal to two, multiplied by one half multiplied by MB multiplied by R squared. Capital R squared, I should say multiplied by the gravitational constant multiplied by MB divided by four RQ when we is equal to the gravitational constant multiplied by MB squared divided by four capital R. So four multiplied by capital R. OK. So the total energy E if the two black holes are separated by a distance of two R would be. And let's remember really quick that E is equal to U plus K. So then we could say that equation nine and this is it, it's separated by a distance of two R. It would be E the energy is equal to negative the gravitational constant multiplied by MB squared divided by two R two multiplied by the capital R plus the gravitational constant multiplied by MB squared divided by four multiplied by capital R. So that means that E is equal to negative, the gravitational constant multiplied by MB squared divided by four multiplied by capital R. So initially, since we are told that the two black holes are very far away from each other, we can state that our approaches infinity. Thus the initial energy will approach zero. So when the two black holes merge, the separation distance is two switch and switch and child radius. Thus, we can write. So let's note that it's two the radius. So two M multiplied by the child radius. So thus, we can write that the final energy ef is equal to minus the gravitational constant multiplied by MB squared divided by or rich and child radius, which gives us equation 10 which states that the delta energy so delta E is equal to ES minus E I which is equal to negative gravitational constant multiplied by MB squared divided by four reconcile radius. OK. So at this stage, we need to express the wretch child radius in terms of MB multiplied by C squared. Also recall that for a black hole, the escape velocity should be greater than or equal to the speed of light. Thus, we need to calculate the escape velocity vesc is equal to C. So the escape velocity is equal to the speed of light when an object is at a distance reen child radius away from the black hole. So we can state that the escape velocity of an object will continue to move away from another body forever. So this can be written as the escape philosophy is equal to the square root of two multiplied by the gravitational constant multiplied by M divided by R. So for the case of this problem, the object will escape a black hole when it is a distance of Gretchen child radius. So let's make a note here up above. So R approaches the Russian style radius. Well, the the escape velocity approaches the speed of light. So we can write that C is equal to the square root of two G, the gravitational constant multiplied by MB divided by the Russians child radius. So now we need to rewrite this equation to solve for the Russian style radius using algebra. So we can call this equation 11. So the Russian style radius is equal to two multiplied by the gravitational constant multiplied by MB all divided by the speed of light squared C squared. So now we need to substitute equation 11 back into equation 10. So when we do that, we get that delta E. So delta E is equal to ES minus E I just equal two, the negative gravitational constant multiplied by MB squared divided by four multiplied by two multiplied by the gravitational constant MB divided by C squared. So that means let's go up here and make a note here. So when we simplify that, that means, and let's write it in green. Our final answer when we simplify will have to be negative MB multiplied by C squared divided by eight. And this is our final answer in terms of MB multiplied by C squared, we did it. So let's go back up to the top to see what, which one of our multiple choice answers is the correct answer. So that means the correct answer has to be the letter D negative MB multiplied by C squared divided by eight. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Verified Solution

Key Concepts

Gravitational Waves

Schwarzschild Radius

Energy-Mass Equivalence (E=mc²)