# Black holes 101(beyond science fiction)

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1 Chapter 13: Gravitation Newton s Law of Gravitation Why is gravity important? Revisit the following: gravitational force, weight, and gravitational energy Stellar motions: the orbits of satellites and celestial objects Black holes 101(beyond science fiction)

2 Introduction Looking at the picture of Saturn, we see a very organized ring around the planet. Why do the particles arrange themselves in such orderly fashion? Gravity is the most important force on the cosmo-level. Why not strong/weak forces? ElectroMagnetic(EM) force also works at large distance? But is it not as important?

3 Newton s Law of Gravitation The gravitational force is always attractive and depends on both the masses of the bodies involved and their separations. F g = Gm 1m 2 r 2 G = N m 2 /kg 2 So small! How was it measured? Everything attracts everything else in the universe

4 Henry Cavendish determines G Gravitational forces were relative until 1798 when Henry Cavendish made the sensitive measurement to determine a numerical value for the constant G.

5 Example of calculating gravitational forces In the three stars shown below, that is the total gravitational force Exerted on the small star at the origin? F 1 = Gm 0 m 1 r 1 2 = N F 1x = F 1y = N = N Gm 0 m 1 2 F 2 r = 2 = r 2 1 F 1 Gm 0 m 2 1 r 2 2 r 1 F 2 = r 2 1 r F 2 1 = 2F 1 = N 2 F x = N F y = N

6 Weight (skip Weight Watchers, just climb upward) Gravity (and hence, weight) decreases as altitude rises. Weight: Total gravitational force from everything else in the universe! In reality, only the nearest steller object matters! w = F g = Gm steller _ objectm 2 R steller _ object g = Gm steller _ object 2 R steller _ object

7 Q12.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.

8 A12.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.

9 Q12.3 Compared to the Earth, Planet X has twice the mass and twice the radius. This means that compared to the Earth s surface gravity, the surface gravity on Planet X is A. 4 times as much. B. twice as much. C. the same. D. 1/2 as much. E. 1/4 as much.

10 A12.3 Compared to the Earth, Planet X has twice the mass and twice the radius. This means that compared to the Earth s surface gravity, the surface gravity on Planet X is A. 4 times as much. B. twice as much. C. the same. D. 1/2 as much. E. 1/4 as much.

11 Gravitational potential energy Objects changing their distance from earth are also changing their potential energy with respect to earth. w grav = r ( Gmm E ) dr = U r 2 U(r) Gmm E r r = U U(r) Gmm E r = U U(r) U(r) = U Gmm E r Is this consistent with U=mgh? In the above plot, where did we choose the point where U=0?

12 Q12.4 Compared to the Earth, Planet X has twice the mass and twice the radius. This means that compared to the amount of energy required to move an object from the Earth s surface to infinity, the amount of energy required to move that same object from Planet X s surface to infinity is A. 4 times as much. B. twice as much. C. the same. D. 1/2 as much. E. 1/4 as much.

13 A12.4 Compared to the Earth, Planet X has twice the mass and twice the radius. This means that compared to the amount of energy required to move an object from the Earth s surface to infinity, the amount of energy required to move that same object from Planet X s surface to infinity is A. 4 times as much. B. twice as much. C. the same. D. 1/2 as much. E. 1/4 as much.

14 Q12.5 A satellite is moving around the Earth in a circular orbit. Over the course of an orbit, the Earth s gravitational force A. does positive work on the satellite. B. does negative work on the satellite. C. does positive work on the satellite during part of the orbit and negative work on the satellite during the other part. D. does zero work on the satellite at all points in the orbit.

15 A12.5 A satellite is moving around the Earth in a circular orbit. Over the course of an orbit, the Earth s gravitational force A. does positive work on the satellite. B. does negative work on the satellite. C. does positive work on the satellite during part of the orbit and negative work on the satellite during the other part. D. does zero work on the satellite at all points in the orbit.

16 What if you are inside the planet g = Boundary conditions g = GM 2 Can you guess what is g between r=0 and r=r? Only the material below r would contribute: The gravitational force at r from the center is only from the mass inside the sphere with radius r (proof at section 12.6*) g r = GM r r 2 = ρ( 4 3 πr3 ) = M Gρ( 4 3 πr3 ) r 2 Assuming the density is constant g r = GM r r 2 = GM( r3 R 3 ) r 2 = GM R 3 r

17 The escape from earth, and the solar system: Minimum v Example 12.5: What is a projectile s velocity when launched, if it could reach n times earth radius (R E ) The escape velocity/speed: escape from gravity and move to an infinitely large distance (does the direction matter?) R 2 =nr E 1 2 mv 2 escape + ( G m Em ) = 0 + ( G m Em ) R E R v escape = 2 Gm E R E 1 2 mv ( G m m E ) = 0 + ( G m m E ) R E nr E v escape = 2(n 1)Gm E nr E n Can you guess what is the escape velocity from the solar system?

18 The escape from earth, and the solar system: Minimum v Example 12.5: What is a projectile s velocity when launched, if it could reach n times earth radius (R E ) The escape velocity/speed: escape from gravity and move to an infinitely large distance (does the direction matter?) R 2 =nr E 1 2 mv 2 escape + ( G m Em ) = 0 + ( G m Em ) R E R v escape = 2 Gm E R E 1 2 mv ( G m m E ) = 0 + ( G m m E ) R E nr E v escape = 2(n 1)Gm E nr E n

19 The escape from earth, and the solar system: Minimum v What is a projectile s velocity when launched, if it could reach n times earth radius (R E ) The escape velocity/speed: escape from gravity and move to an infinitely large distance (does the direction matter?) R 2 =nr E 1 2 mv 2 escape + ( G m Em ) = 0 + ( G m Em ) R E R v escape = 2 Gm E R E 1 2 mv ( G m m E ) = 0 + ( G m m E ) R E nr E v escape = 2(n 1)Gm E nr E n sun v escape = 2 Gm sun R 0

20 Satellite motion Do satellites always move in circles? G m m E = m v 2 r 2 r v = T =? G m E r K =?U =? E =? For every radius, there is only one speed that will ensure the circular motion, independent of satellite mass

21 Synchronous satellite Synchronous satellite: orbiting the earth at the same rate, direction as the earth rotation; (T=24hrs, m E =5.97x10 24 kg, R E =6380km) 3 How high should it be from ground? T = 2πr 2 Gm E r = ( T 2 1 Gm E ) 4π 2 3 Δr = r R E How much work needs to be done to move the 1000kg satellite into orbit? Is it just the increase of the potential energy? w = ΔU + ΔK ΔU = Gmm E r ( Gmm E R E ) ΔK = 1 2 mv 2 = 1 2 m Gm E r

22 Kepler s laws for planetary motion:first law Planet/Wanderer: We are not the center of the world Kepler s first law: Each planet moves in an elliptical orbit with the sun at one focus. Why is it not always a circle Extreme cases: If the planet has a very large speed, or it has just the right speed. In almost all cases, the actual speed is somewhere in between

23 Kepler s laws for planetary motion:second law Kepler s second law: A line from the sun to a given planet sweeps out equal areas in equal times What does this mean? Hint: What is the angular momentum of the planet? 1 da dt = 2 r2 dθ = 1 dθ r(r dt 2 dt ) = 1 2 rv = 1 rv sinφ 2 L = r(mv)sinφ Apparently, the planet s angular momentum doesn t change. Why?

24 Kepler s laws for planetary motion:second law Kepler s second law: A line from the sun to a given planet sweeps out equal areas in equal times What does this mean? Hint: What is the angular momentum of the planet? 1 da dt = 2 r2 dθ = 1 dθ r(r dt 2 dt ) = 1 2 rv = 1 rv sinφ 2 L = r(mv)sinφ dl Apparently, the planet s angular dt = τ = r F = 0 momentum doesn t change. Why? r, and F are in the same line!

25 Q12.6 A planet (P) is moving around the Sun (S) in an elliptical orbit. As the planet moves from aphelion to perihelion, the Sun s gravitational force A. does positive work on the planet. B. does negative work on the planet. C. does positive work on the planet during part of the motion and negative work during the other part. D. does zero work on the planet at all points between aphelion and perihelion.

26 A12.6 A planet (P) is moving around the Sun (S) in an elliptical orbit. As the planet moves from aphelion to perihelion, the Sun s gravitational force A. does positive work on the planet. B. does negative work on the planet. C. does positive work on the planet during part of the motion and negative work during the other part. D. does zero work on the planet at all points between aphelion and perihelion.

27 Q12.7 A planet (P) is moving around the Sun (S) in an elliptical orbit. As the planet moves from aphelion to perihelion, the planet s angular momentum A. increases during part of the motion and decreases during the rest of the motion. B. increases at all times. C. decreases at all times. D. remains the same at all times.

28 A12.7 A planet (P) is moving around the Sun (S) in an elliptical orbit. As the planet moves from aphelion to perihelion, the planet s angular momentum A. increases during part of the motion and decreases during the rest of the motion. B. increases at all times. C. decreases at all times. D. remains the same at all times.

29 Kepler s laws for planetary motion:third law Kepler s third law: The period of the planets are proportional to the 3/2 powers of the major axis lengths of their orbits T = 2πa 3 2 Gm E We have already known this from the synchronous satellite (circular Orbit). Why would this still be true if the orbit is elliptical? First and third law: Only valid when F g 1 r 2 Second law: As long as gravity is a central force How were the distances between planets and stars measured then and now?

30 Halley s comet Why is Comet Halley so famous (Is it because it is the brightest)? Knowing the comet return every 75.5 years (2.38x10 9 s), and the sun s mass is 1.99x10 30 kg, Can you deduce the semi-major axis of its orbit T = 2πa 3 2 Gm S a = ( T 2 1 Gm S ) 4π 2 3

31 Black hole 101 What is a black hole? A black hole is formed when a star collapses due to its own gravity, if the internal pressure could no longer be maintained via nuclear reactions (fuel runs out) Black hole s gravity is so large that even light can not escape Try escape velocity being speed of light, and the minimum mass is three times that of the sun: 3x(1.99x10 30 )kg Event horizon v escape c = 2 Gm B R B R B = 2Gm B = 2G(3m S ) c 2 c 2 Wrong equation, Correct result Atoms sometimes are just like a solar system, with electrons orbiting protons/neutrons, and lots of empty space in between? What happens the atoms collapse and there is no more empty space between the nucleons?

32 How to detect black holes If light cannot escape black holes, how do we know we are there? X-ray sources Orbits of surrounding stars

33 Summary Newton s law of gravitation: F g = Gm 1m 2 r 2 G = N m 2 /kg 2 Weight and gravitational potential energy w = F g = Gm m steller _ object 2 R steller _ object g = Gm steller _ object 2 R steller _ object Compare this with U=mgh U = Gm steller _ object R steller _ object Satellite in circular orbit: v = G m E r T = 2πr 3 2 Gm E When dealing with steller objects, satellite, we DON t use U=mgh! Kepler s laws: Second law angular momentum conservation

### A. 81 2 = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.

Q12.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2

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