Sec. 6.3 Gravitational Energy

Transcription

1 Sec. 6.3 Gravitational Energy Learning Goal: By the end of today, I will be able to determine the change in potential energy between two objects with mass AND I will be able to determine the escape velocity of an object in orbit. There is a gravitational potential energy between any two masses. If the distance between the masses changes, so does the potential energy, similar to a spring system work must be done. Apr 8 3:07 PM Apr 8 3:09 PM 1

2 Proof is on page Area under the curve ~ height x base ~ (geometric ave) x (change in r) Arithmetic Average (add together divide by two) Geometric Average (multiply together and square root) constants vertical stretches parent curve multiply together square root both terms and simplify distribute cancel r's rearrange order of terms change positive sign to subtracting a negative should always be positive if you are moving away or increasing the distance delta E g = E g2 E g1 delta E g = E g (r 2 ) E g (r 1 ) functional notation Apr 8 3:09 PM Apr 8 9:14 PM 2

3 Near Earth's Surface As r 2 gets very, very large, E 2 goes to zero. As an object moves out of gravitational range, the change in potential energy can be simplified to the above. (This idea will show up again in particle discussions.) As you are moving away, or higher, delta Eg is positive Apr 8 3:14 PM Apr 8 3:39 PM 3

4 Example What is the change in gravitational potential energy of a 64.5 kg astronaut, lifted from Earth s surface into a circular orbit of altitude 4.40 x 10 2 km? Escaping a Gravitational Field The negative value of this potential energy is characteristic of a potential well,a name derived from the shape of the graph of the gravitational potential energy as a function of separation distance General Eg function Just as things fall into wells in real life, so do space objects. If an object doesn't have enough kinetic energy, it can fall into the potential well around a large massive object. Even though the values for the astronaut s gravitational potential energy are negative at both positions, the change in Eg, when the astronaut s distance from Earth increases, is positive,indicating an increase in gravitational potential energy. Apr 8 4:21 PM Apr 8 9:29 PM 4

5 To escape Earth's potential well, the kinetic energy must be more than the E g highlighted on the graph. zero escape speed the minimum speed needed to project a mass m from the surface of mass M to just escape the gravitational force of M E T must be equal or greater than zero, therefore E k must be equal or greater than E g escape energy the minimum kinetic energy needed to project a mass m from the surface of mass M to just escape the gravitational force of M E T = E g + E k E k To just escape, E T = 0, E k + E g = 0 E k = E g On Earth E g As you move farther from Earth, the potential well gets smaller, requiring a smaller velocity to escape. binding energy the amount of additional kinetic energy needed by a mass m to just escape from a mass M Note a rocket flying away from Earth is different than a satellite orbiting Earth. "Escape Speed" Apr 8 9:55 PM Apr 8 10:06 PM 5

6 Note, E T can be a negative value. Wow! Apr 8 10:08 PM Apr 8 10:22 PM 6

7 Three Possible Outcomes As you move farther from Earth, the potential well gets smaller, requiring a smaller velocity to escape. Apr 8 10:26 PM Apr 8 11:09 PM 7

8 a black hole a very dense body in space with a gravitational field so strong that nothing can escape from it event horizon the surface of a blackhole singularity the dense centre of a blackhole Schwartzschild radius the distance from the centre of the singularity to the event horizon Event Horizon You can't see me! singularity radius First Black Hole Discovery In 1972, Professor Tom Bolton, while working at the University of Toronto s David Dunlap Observatory in Richmond Hill, Ontario, was investigating a point in space, Cygnus X 1, because it was a source of Xrays. It turned out to be one of the most significant discoveries in astronomy: a black hole. This was the first evidence to support the existence of black holes, which were previously hypothetical objects. Apr 8 11:12 PM Apr 8 11:12 PM 8

9 Homework Read page page 287 #1, 2, 4 page 293 #8, 9, 10 page 294 #1, 3, 4, 6 Apr 8 11:17 PM Apr 8 3:39 PM 9