Mechanics Beyond the Earth

1. Mechanics Beyond the Earth Ptolemy (0085 – 0165): Formalized and quantified the ancient-Greek earth-centered world-view that accounted for “retrograde” motion of the planets by supposing that these “wanderers” actually moved in orbits containing “epicycles”—i.e. smaller circles upon larger circles. Copernicus (1473 – 1543): Replaced the earth-centered model with a sun-centered model, which accounted for planetary observations without the need to assume such “epicycles.” Brahe (1546 – 1601): Meticulous astronomer whose life work revealed (to Kepler) that neither Ptolemy nor Copernicus had quite an accurate model. Kepler (1571 – 1630): Showed with math that Brahe’s data fit Copernicus’ basic idea, but with elliptical orbits; suggested inter-body force. Galileo (1564 – 1642): First to observe other planet’s (Jupiter’s) moons directly (via telescope), thus undermining the earth-centered world view: Not all bodies orbit the earth. Newton (1642 – 1727): Showed with math that Kepler’s elliptical orbits indeed result from a pulling force—the same that makes earth-bound objects fall. OSU PH 212, Before Class #28

2. Ideas that Newton might have pondered… The Moon stays in orbit around the Earth—never crashing down on us—because…? It is somehow beyond the pull of Earth’s gravity? It is being pulled by the Sun and planets, as well as by Earth, so the net force keeps it in orbit? There are other, unknown forces that keep it in orbit? OSU PH 212, Before Class #28

3. The Universal Law of Gravitation FG = mg. But what determines our local g value? And how could a force that can send apples straight to the ground be the same force that merely holds the moon at a (very far) distance? Isaac Newton looked at data compiled by earlier astronomers and made the powerful mathematical connection between ordinary projectiles here on Earth and the motions of the moon and the planets and sun: The gravitational force of attraction between any two masses, m1 and m2, is given by FG = Gm1m2/r2, where r is the distance between the centers of the two masses, and G (= 6.67 x 10-11 N·m2/kg2) is a universal constant. OSU PH 212, Before Class #28

4. Two objects are not in contact, but they pull on each other via gravity, and no other forces act on either object.Which statement is always true? • The gravitational force magnitudes are equal. • The gravitational force directions are equal. • The gravitational acceleration magnitudes are equal. • All of the above. • None of the above. OSU PH 212, Before Class #28

5. Two objects are not in contact, but they pull on each other via gravity, and no other forces act on either object.Which statement is always true? • The gravitational force magnitudes are equal. • The gravitational force directions are equal. • The gravitational acceleration magnitudes are equal. • All of the above. • None of the above. OSU PH 212, Before Class #28

6. Example: Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the force from the earth acting on B to that of the earth acting on A? FG.EB/FG.EA = [GmEmB/rB2]/[GmEmA/rA2] = [rA/rB]2 = 1/4 OSU PH 212, Before Class #28

7. Notice what Newton’s Law of Universal Gravitation now implies for the meaning or “function” of mass. Before, we regarded mass as merely the measure of an object’s inertia —its “resistance to acceleration.”Now it’s also a measure of its ability to attract other masses.Notice, too, what Newton’s Law of Universal Gravitation implies for our local (earth-surface) acceleration due to gravity. Consider any object of mass mobj…FG.Eobj = mobjg = GMEmobj/robj2Therefore: gsurface = GME/Rearth2 OSU PH 212, Before Class #28

8. The Magnitude of the Acceleration due to Gravity So, why do we use an earth-surface value of g = 9.80 m/s2, rather than the above value (9.83)? Because the earth is rotating. See page 192 in Chapter 8 for that discussion. OSU PH 212, Before Class #28

9. Simple Models of Circular Orbits Gravitational – Satellites Electrical – The Bohr model of hydrogen Magnetic – ions in mass spectrometers OSU PH 212, Before Class #28

10. When Gravity is the Sole Cause of Radial Acceleration It is entirely possible for an object to be in a state of free-fall motion toward a planet and yet the object never reaches the planet. How is this so? Consider the world’s finest baseball pitcher (and for the moment, disregard the problem of air drag).... What if the ball were thrown with enough speed so that as it falls toward the center of the Earth, it has also moved horizontally enough so that its “fall” path matches the curvature of the Earth? The planet’s gravity exerts the net force that keeps the object in a circular path—and the planet’s shape keeps it “out of the way” of that circular path! OSU PH 212, Before Class #28

11. This concept was the key for Isaac Newton. This explains why the moon can be falling for millions of years toward Earth, yet never reach it: The force of gravity supplies just exactly the right magnitude and direction to act as a radial force on the moon (or any other terrestrial satellite). In other words: GmEm/r2 = mv2/r If we rearrange this to solve for v, we can predict the speed v necessary for an orbit of any given radius (and notice that that it does not depend on the mass of the object—only the mass of the planet): vorbit = [GmE/r]1/2 [So: With what horizontal speed would the pitcher need to throw a baseball at 1 m above the surface of the earth so that—without air resistance—it would never hit the Earth? About 7900 m/s!] OSU PH 212, Before Class #28