1. Gravitation Using the law of universal gravitation to gain perspective on planets, moons, stars and black holes.

2. What determines the force of Gravity? Any two objects will exert an attractive force on one another based on the following: – Their masses and the distance separating them Planets and stars have gravitational fields or regions surrounding them where objects accelerate at various rates. The bigger the mass of the planet or star and the closer to it the stronger the field.

3. Newton’s Law of Universal Gravitation: Where: F G = force of gravity (N) G = Universal gravitational constant = 6.67x10 -11 Nm 2 /kg 2 m 1 and m 2 = masses of objects 1 and 2 r = radius = distance separating objects (m)

4. Gravitational field strength decreases with distance from the surface. F g F g

5. Gravitational PE becomes increasingly negative as objects get closer to the surface. It is 0 infinitely far away.

6. Escape Velocity This is the minimum velocity an object must have to completely escape from the gravitational field of a body. V esc

7. How to Find Escape Velocity Use Energy: If the kinetic energy of an object on the surface of the planet equals the negative of the gravitational PE at that point then the object is moving at its escape velocity and will break free of the gravitational pull of the planet. (Assuming no energy lost to other forms) Where: V esc = escape velocity (m/s) G = universal gravitational constant =6.67x10 -11 Nm 2 /kg 2 m co = mass of central object (kg) r co = radius of central object (m)

8. Sample Escape Velocities: Moon 2,380 m/s Earth 11,200 m/s Jupiter 59,500 m/s Sun 618,000 m/s

9. The escape velocity of a black hole is the speed of light! Black holes are so massive that their gravitational field will not allow light to exit from it’s event horizon. At the event horizon the escape velocity is equal to the speed of light (c = 2.998*10 8 m/s). Vesc = c

10. Using “goalseek” to find the radius of a black hole given its mass: 1.Enter the mass of the black hole in the mass cell (C2) 2.Open goal seek (in the tools menu or “What if analysis”). 3.Set the cell (E6) for escape velocity equal to 2.998e8 by changing the central object radius (D2). 4.Press enter and the radius of the event horizon will be left in cell D2.

11. Tips: The increment of the position is 1/5 th the radius of the central object (cell D2). Acceleration due to gravity, weight and PE g on the surface of the central object can be found in the first nonzero row (row 10). Use the following notation: Either “=6.67*10^-11” or “6.67e-11” to enter numbers in scientific notation.