Physics 201: Lecture 25, Pg 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.

Physics 201: Lecture 25, Pg 2 Exam results l Mean/Median 59/100 l Nominal exam curve A AB B BC C D F below 25

Physics 201: Lecture 25, Pg 3 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 Newton’s Universal “Law” of Gravity

Physics 201: Lecture 25, Pg 4 Newton’s Universal “Law” of Gravity “Big” G is the Universal Gravitational Constant G = x Nm 2 / kg 2 Two 1 Kg particles, 1 meter away  F g = 6.67 x N (About 39 orders of magnitude weaker than the electromagnetic force.) The force points along the line connecting the two objects.

Physics 201: Lecture 25, Pg 5 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 Little g and Big G where we have used a local constant acceleration: On Earth, near sea level, it can be shown that g surface ≈ 9.8 m/s 2.

Physics 201: Lecture 25, Pg 6 g varies with altitude l With respect to the Earth’s surface

Physics 201: Lecture 25, Pg 7 At what distance are the Earth’s and moon’s gravitational forces equal l Earth to moon distance = 3.8 x 10 8 m l Earth’s mass = 6.0 x kg l Moon’s mass = 7.4 x kg

Physics 201: Lecture 25, Pg 8 The gravitational “field” l 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. l The presence of a Gravitational Field is indicated by a field vector representation with both a magnitude and a direction. M

Physics 201: Lecture 25, Pg 9 Compare and contrast l Old representation M l New representation

Physics 201: Lecture 25, Pg 10 When two isolated masses m 1 and m 2 interact over large distances, they have a gravitational potential energy of Gravitational Potential Energy The “zero” of potential energy occurs at r = ∞, where the force goes to zero. Note that this equation gives the potential energy of masses m 1 and m 2 when their centers are separated by a distance r. Recall for 1D:

Physics 201: Lecture 25, Pg 11 A plot of the gravitational potential energy l U g = 0 at r = ∞ l We use infinity as reference point for U g (∞)= 0 l This very different than l U(r) = mg(r-r 0 ) l Referencing infinity has some clear advantages l All r allow for stable orbits but if the total mechanical energy, K+ U > 0, then the object will “escape”.

Physics 201: Lecture 25, Pg 12 It is the same physics l Shifting the reference point to R E.

Physics 201: Lecture 25, Pg 13 Gravitational Potential Energy with many masses l If there are multiple particles l Neglects the potential energy of the many mass elements that make up m 1, m 2 & m 3 (aka the “self energy”

Physics 201: Lecture 25, Pg 14 Dynamics of satellites in circular orbits Circular orbit of mass m and radius r l Force: F G = G Mm/r 2 =mv 2 /r l Speed: v 2 = G M/r (independent of m) l Kinetic energy: K = ½ mv 2 = ½ GMm/r l Potential energy U G = - GMm/r U G = -2 K This is a general result l Total Mechanical Energy: l E = K + U G = ½ GMm/r -GMm/r E = - ½ GMm/r

Physics 201: Lecture 25, Pg 15 Changing orbits l With man-made objects we need to change the orbit Examples: 1. Low Earth orbit to geosynchronous orbit 2. Achieve escape velocity l 1. We must increase the potential energy but one can decrease the kinetic energy consistent with

Physics 201: Lecture 25, Pg 16 Changing orbits l Low Earth orbit to geosynchronous orbit l A 470 kg satellite, initially at a low Earth orbit of h i = 280 km is to be boosted to geosynchronous orbit h f = km l What is the minimum energy required to do so? (r E = 6400 km, M = 6 x kg

Physics 201: Lecture 25, Pg 17 Escaping Earth orbit l 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)

Physics 201: Lecture 25, Pg 18 Escaping Earth orbit l 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)

Physics 201: Lecture 25, Pg 19 Escaping Earth orbit l 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

Physics 201: Lecture 25, Pg 20 Some interesting numbers l First Astronautical Speed  Low Earth orbit: v = 7.9 km/s l Second Astronautical Speed  Escaping the Earth: v = 11 km/s l Third Astronautical Speed  Escaping Solar system: v= 42 km/s

Physics 201: Lecture 25, Pg 21 Next time l Chapter 14 sections 1 to 3, Fluids