Syll. State.: 9.2.1-9.2.9—due Friday, October 3 Gravitation Reading: pp. 124-129; 133-137 Syll. State.: 9.2.1-9.2.9—due Friday, October 3

Newton’s Law of Universal Gravitation “Every material particle in the Universe attracts every other material particle with a force that is directly proportional to the product of the masses of the particles and that is inversely proportional to the square of the distance between them.” Principia, Sir Isaac Newton

Gravitation The gravitational forces obey Newton’s 3rd law of motion—they are a pair of action/reaction forces EVERY particle in the universe obeys this law

Gravitation F12 = strength of gravitational force body 1 exerts on body 2 F21 = strength of gravitational force body 2 exerts on body 1 m1 and m2 = masses of the two bodies r = separation distance between the center of mass for each of the bodies G = Universal Gravitation constant = 6.67 x 10-11 N m2 kg-2

Sample Problem #1 Find the force between the sun and the earth.

Me Re m If a spherical object is uniform in mass, we can safely assume that the gravitational force due to this mass is exactly the same as if all the mass were concentrated at the very center. Therefore, for something at the surface of this mass, we can deduce that the following is true: Thus we can conclude that the acceleration due to the gravity of a uniform mass is related to the radius of the large mass (in this example, Earth…)

Sample problem #2 Determine the acceleration due to gravity at the surface of a planet that is 10 times more massive than the earth and has a radius 20 times larger.

Gravitational Field Strength Universal gravitation is VERY similar to Coulomb’s Law (review electrostatics!), except there is no such thing as negative mass… Therefore, Gravitational forces will always be attractive forces. The gravitational field is a “field of influence radiating outward from a particle of mass m” Gravitational Field Strength at a given point is the force exerted per unit mass on a particle of small mass placed at that point. The strength of the gravitational field is essentially equal to the acceleration of the particle (g) Units for the field strength, although they are equal to m s-2, are typically listed as N kg-1

Gravitational Field (diagram) Gravitational field lines are vectors that always are directed radially toward the center of the mass. (remember…the field lines (vectors) will indicate the direction a smaller mass will be pulled as a result of the gravitational force caused by the field…)

Gravitational Potential Energy Energy that is “stored” in an object’s gravitational field. Work must be done in order to change the position of a second mass from an infinitely far away point in space to a position relatively near the first mass. Remember: Work-Energy Theorem The amount of work done on an object is equal to the change in energy of that object. In this case, the energy change is all potential.

Gravitational Potential Energy Work is done when moving the mass from a distance infinitely far from the source of a gravitational field to some distance R from the source This work is stored in the gravitational field between the two masses: Gravitational Potential Energy

Gravitational Potential (V) A field, defined at every point in space, but a scalar quantity Defined as: The work done per unit mass to bring a small point mass (m) from infinity to a point P.

Total Energy of an Object in Orbit Total energy, as usual, is the sum of the orbiting object’s gravitational potential and kinetic energies: However, we can simplify this…

Total Energy… simplified! Solve for v2 using Newton’s laws: Substitute into energy equation:

Graphing Mechanical Energy: Graph of gravitational potential energy as a function of distance from a planet’s surface

Orbital Motion What kind of force causes objects to move along a circular path? Centripetal Force! To determine the orbital speed of a mass that is orbiting around a larger central mass, we must first recognize that the centripetal force keeping it in orbit is actually the gravitational force, so…

Orbital Motion And… So it follows that: The orbital speed of a satellite, therefore, can be determined by using:

Orbital Motion The previous example shows us how the orbital speed of a satellite depends on its orbital radius. The farther out a satellite is from Earth’s surface, the slower it can go and still remain in orbit How far out (what orbital radius) must a satellite have to remain in geosynchronous orbit?

Kepler’s laws of Planetary Motion 1st Law: Planets move in ellipses around the sun, the sun being at one of the foci of each ellipse. 2nd Law: The line from the sun to he planet sweeps out equal areas in equal times. 3rd Law: The time taken to complete one full revolution is proportional to the 3/2 power of the mean distance from the sun. Or: the orbital period squared is directly proportional to the cube of the mean distance from the sun.

Orbital Period, related to Kepler’s third law of planetary motion… Assume a circular orbit…the average orbital speed is equal to the distance traveled by the planet or satellite in one orbital period: Which Means… Using the relationship we previously deduced: We can relate Newton’s law of gravitation to Kepler’s 3rd law of planetary motion (see problem #7 from your homework assignment)

Weightlessness The forces that act on an astronaut (or anything else) in orbit in a spacecraft, some distance r from the center of the earth are: His weight (gravitational force from Earth) The normal force (N) from the “floor” of the spacecraft. What is the net force acting on the astronaut?

r2 r r2 r N = (m/r){(GM/r) – (v2)} Fnet = (GMm) – N = mv2 (GMm) – mv2 = N r2 r N = (m/r){(GM/r) – (v2)} When we looked at Orbital motion, we showed that (GM/r) = v2 …so our equation above mathematically shows that the Normal force that the astronaut feels is equal to 0 N... Therefore: “Weightlessness” implies not that there is no gravity, but rather that there is no reaction force exerting the upward force that we sense as our weight.