Gravitation Reading: pp

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.” “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 Principia, Sir Isaac Newton

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

Gravitation F 12 = strength of gravitational force body 1 exerts on body 2 F 12 = strength of gravitational force body 1 exerts on body 2 F 21 = strength of gravitational force body 2 exerts on body 1 F 21 = strength of gravitational force body 2 exerts on body 1 m 1 and m 2 = masses of the two bodies m 1 and m 2 = masses of the two bodies r = separation distance between the center of mass for each of the bodies r = separation distance between the center of mass for each of the bodies G = Universal Gravitation constant G = Universal Gravitation constant = 6.67 x N m 2 kg -2 = 6.67 x N m 2 kg -2

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

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. 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: 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…) 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…) MeMe ReRe m

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. 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.

Orbital Motion What kind of force causes objects to move along a circular path? 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… 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… And… So it follows that: So it follows that: The orbital speed of a satellite, therefore, can be determined by using: 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 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 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? How far out (what orbital radius) must a satellite have to remain in geosynchronous orbit?

Gravitational Field Strength, g The gravitational field is a “field of influence radiating outward from a particle of mass, M” Gravitational Field Strength at a given point is equivalent to the force exerted per unit mass acting on a small mass, m, placed at that point. Units  N·kg -1

Gravitational Field (diagram) Gravitational field lines are vectors that always are directed radially toward the center of the mass. 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…)

Nature of Science: think about this… What happens to the gravitational field strength as you travel deeper into the earth? What happens to the gravitational field strength as you travel deeper into the earth? What does field strength depend on? What does field strength depend on? What assumptions must we make? What assumptions must we make?

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