1 / 21

Lecture 25

Lecture 25. Jupiter and 4 of its moons under the influence of gravity Goal: To use Newton’s theory of gravity to analyze the motion of satellites and planets. Exam results. Mean/Median 59/100 Nominal exam curve A 81-100 AB 71-80 B 61-70 BC 51-60 C 31-50 D 25-30

clem
Download Presentation

Lecture 25

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 25 Jupiter and 4 of its moons under the influence of gravity Goal: To use Newton’s theory of gravity to analyze the motion of satellites and planets.

  2. Exam results • Mean/Median 59/100 • Nominal exam curve A 81-100 AB 71-80 B 61-70 BC 51-60 C 31-50 D 25-30 F below 25

  3. Newton’s Universal “Law” of Gravity Newton proposed that every object in the universe attracts every other object. The Law: Any pair of objects in the Universe attract each other with a force that is proportional to the products of their masses and inversely proportional to the square of their distance

  4. Newton’s Universal “Law” of Gravity “Big” G is the Universal Gravitational Constant G = 6.673 x 10-11 Nm2 / kg2 Two 1 Kg particles, 1 meter away Fg= 6.67 x 10-11 N (About 39 orders of magnitude weaker than the electromagnetic force.) The force points along the line connecting the two objects.

  5. Little g and Big G Suppose an object of mass m is on the surface of a planet of mass M and radius R. The local gravitational force may be written as where we have used a local constant acceleration: On Earth, near sea level, it can be shown that gsurface ≈ 9.8 m/s2.

  6. g varies with altitude • With respect to the Earth’s surface

  7. At what distance are the Earth’s and moon’s gravitational forces equal • Earth to moon distance = 3.8 x 108 m • Earth’s mass = 6.0 x 1024 kg • Moon’s mass = 7.4 x 1022 kg

  8. The gravitational “field” • To quantify this invisible force (action at a distance) even when there is no second mass we introduce a construct called a gravitational “force” field. • The presence of a Gravitational Field is indicated by a field vector representation with both a magnitude and a direction. M

  9. Compare and contrast • Old representation • New representation M

  10. Gravitational Potential Energy Recall for 1D: When two isolated masses m1 and m2 interact over large distances, they have a gravitational potential energy of The “zero” of potential energy occurs at r = ∞, where the force goes to zero. Note that this equation gives the potential energy of masses m1 and m2 when their centers are separated by a distance r.

  11. A plot of the gravitational potential energy • Ug= 0 at r = ∞ • We use infinity as reference point for Ug(∞)= 0 • This very different than • U(r) = mg(r-r0) • Referencing infinity has some clear advantages • All r allow for stable orbits but if the total mechanical energy, K+ U > 0, then the object will “escape”.

  12. It is the same physics • Shifting the reference point to RE.

  13. Gravitational Potential Energy with many masses • If there are multiple particles • Neglects the potential energy of the many mass elements that make up m1, m2 & m3 (aka the “self energy”

  14. Dynamics of satellites in circular orbits Circular orbit of mass m and radius r • Force: FG= G Mm/r2 =mv2/r • Speed: v2 = G M/r (independent of m) • Kinetic energy: K = ½ mv2 = ½ GMm/r • Potential energy UG = - GMm/r UG = -2 K This is a general result • Total Mechanical Energy: • E = K + UG = ½ GMm/r -GMm/r E = - ½ GMm/r

  15. Changing orbits • With man-made objects we need to change the orbit Examples: 1. Low Earth orbit to geosynchronous orbit 2. Achieve escape velocity • 1. We must increase the potential energy but one can decrease the kinetic energy consistent with

  16. Changing orbits • Low Earth orbit to geosynchronous orbit • A 470 kg satellite, initially at a low Earth orbit of hi = 280 km is to be boosted to geosynchronous orbit hf = 35800 km • What is the minimum energy required to do so? (rE = 6400 km, M = 6 x 1024 kg

  17. Escaping Earth orbit • Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation)

  18. Escaping Earth orbit • Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation)

  19. Escaping Earth orbit • Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation) implies infinite height

  20. Some interesting numbers • First Astronautical Speed • Low Earth orbit: v = 7.9 km/s • Second Astronautical Speed • Escaping the Earth: v = 11 km/s • Third Astronautical Speed • Escaping Solar system: v= 42 km/s

  21. Next time • Chapter 14 sections 1 to 3, Fluids

More Related