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If you where to shrink the Earth and put all of its mass into a small enough radius, you could form a black hole with mass equal to the mass of the Earth. Calculate the radius at which the Earth would become a black hole. Answer this problem in the following two questions below. Question 5 First you need find an equation for the escape velocity and then solve for the radius r. The escape velocity is that at which the Kinetic Energy KE = mv² is equal to the Potential Energy PE = mgr (notice that you are used to see PE - mgh, since we usually measure potential energy relative to the surface of the earth, now we want to find the potential at the surface of the earth, soh =r). Notice though that the gravitational acceleration g is not constant if you change the radius r, we can find how g depends on r using Newton's Law of Gravitation and Newton's 2nd Law: g = = GmM = GA Set KE - PE and solve for r, what do you get? HINT: Remember to set g = CM Or = 2GM Or= M Question 6 Now that you have an equation for r, set the escape velocity to the speed of light, i.e. v =c (that is how we define a black hole), and the mass M to the mass of the Earth. Use the values below in the equation and calculate r. C-3 x 10 m/s Gravitational Constant 6.7 x 1011 m kg s2 Mass of the Earth 6 x 10^24 kg Give your answer in units of mm (millimeters)

If you where to shrink the Earth and put all of its mass into a small enough radius, you could form a black hole with mass equal to the mass of the Earth. Calculate the radius at which the Earth would become a black hole. Answer this problem in the following two questions below. Question 5 First you need find an equation for the escape velocity and then solve for the radius r. The escape velocity is that at which the Kinetic Energy KE = mv² is equal to the Potential Energy PE = mgr (notice that you are used to see PE - mgh, since we usually measure potential energy relative to the surface of the earth, now we want to find the potential at the surface of the earth, soh =r). Notice though that the gravitational acceleration g is not constant if you change the radius r, we can find how g depends on r using Newton's Law of Gravitation and Newton's 2nd Law: g = = GmM = GA Set KE - PE and solve for r, what do you get? HINT: Remember to set g = CM Or = 2GM Or= M Question 6 Now that you have an equation for r, set the escape velocity to the speed of light, i.e. v =c (that is how we define a black hole), and the mass M to the mass of the Earth. Use the values below in the equation and calculate r. C-3 x 10 m/s Gravitational Constant 6.7 x 1011 m kg s2 Mass of the Earth 6 x 10^24 kg Give your answer in units of mm (millimeters)

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If you where to shrink the Earth and put all of its mass into a small enough radius, you could form a black hole with mass equal to the mass of the Earth. Calculate the radius at which the Earth would become a black hole. Answer this problem in the following two questions below. Question 5 First you need find an equation for the escape velocity and then solve for the radius r. The escape velocity is that at which the Kinetic Energy KE = mv² is equal to the Potential Energy PE = mgr (notice that you are used to see PE = mgh, since we usually measure potential energy relative to the surface of the earth, now we want to find the potential at the surface of the earth, so h = r). F. GmM Notice though that the gravitational accelerationg is not constant if you change the radius r, we can find how g depends on r using Newton's Law of Gravitation and Newton's 2nd Law: g = GM r2 m mr2 Set KE = PE and solve for r, what do you get? GM HINT: Remember to set g = Or= 2GM Or = GM O r = Or 2GM Question 6 Now that you have an equation for r, set the escape velocity to the speed of light, i.e. v = c (that is how we define a black hole), and the mass M to the mass of the Earth. Use the values below in the equation and calculate r. c = 3 x 108 m/s Gravitational Constant = 6.7 × 1011 m³ kg 1 s-2 Mass of the Earth = 6 x 10^24 kg Give your answer in units of mm (millimeters)
Transcribed Image Text:If you where to shrink the Earth and put all of its mass into a small enough radius, you could form a black hole with mass equal to the mass of the Earth. Calculate the radius at which the Earth would become a black hole. Answer this problem in the following two questions below. Question 5 First you need find an equation for the escape velocity and then solve for the radius r. The escape velocity is that at which the Kinetic Energy KE = mv² is equal to the Potential Energy PE = mgr (notice that you are used to see PE = mgh, since we usually measure potential energy relative to the surface of the earth, now we want to find the potential at the surface of the earth, so h = r). F. GmM Notice though that the gravitational accelerationg is not constant if you change the radius r, we can find how g depends on r using Newton's Law of Gravitation and Newton's 2nd Law: g = GM r2 m mr2 Set KE = PE and solve for r, what do you get? GM HINT: Remember to set g = Or= 2GM Or = GM O r = Or 2GM Question 6 Now that you have an equation for r, set the escape velocity to the speed of light, i.e. v = c (that is how we define a black hole), and the mass M to the mass of the Earth. Use the values below in the equation and calculate r. c = 3 x 108 m/s Gravitational Constant = 6.7 × 1011 m³ kg 1 s-2 Mass of the Earth = 6 x 10^24 kg Give your answer in units of mm (millimeters)
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