Law of gravitation Kepler’s laws Energy in planetary motion Atomic spectra and the Bohr model Orbital motion (chapter eleven)

Newton’s law of universal gravitation, revisited Newton postulated an inverse square law for the gravitational force between any two masses He also showed that the gravitational force exerted by a sphere is equal to that of a point mass at the center, equal to the enclosed mass only.

Gravitational force Example: satellites Speed is independent of mass – depends only on radius (height) of orbit.

Structural models Models that describe a system which generally cannot be altered by humans. Often contains the following features: Physical components Location of components Time evolution Agreement between predictions and observations, and even predictions of not-yet- observed phenomena

Kepler’s laws First law: Each planet moves in an elliptical orbit a b F 1, F 2 : foci a: semimajor axis b: semiminor axis e=c/a: eccentricity r2r2 r1r1 F1F1 F2F2 c eccentricity of Earth’s orbit = 0.017, Pluto’s orbit = 0.25 Ellipses: only types of bound orbits for inverse square force

Kepler’s laws Second law: The radius vector from the sun to any planet sweeps out equal area in equal time intervals. Since the force is directed along the line between the two masses, the cross product is identically zero Therefore, the angular momentum of the planet is constant

Kepler’s laws Second law dA Using this result to find the area swept out as a function of time drdr Integrating this gives the same result for identical time intervals

Kepler’s laws Third law The square of the orbital period is proportional to the cube of the semimajor axis of the orbit Easiest to demonstrate with circular orbits: For an elliptical orbit with semimajor axis a

Energy in planetary motion The sum of the kinetic and potential energies of a mass in the gravitational field of a much larger mass is The potential is defined for the case of U g  0 as r . In principle, we can add any constant to it and it will still be valid, since only the change in energy is important. We can use Newton’s 2 nd law to find one other relation

Energy in planetary motion So the total energy is just For elliptical orbits, r  a

Escape speed An object with some speed at the surface of the earth may have enough energy to escape the earth’s gravitational potential. First, suing conservation of energy, consider the maximum height a mass with speed v could reach This gives a general relation for the speed needed to reach a height r max :

Escape speed Setting r max to infinity defines the escape speed of an object Independent of both the direction of v and the mass of the object. Earth: v esc = 11.2 km/s Moon: v esc = 2.3 km/s

Black holes If the escape speed of a mass equals the speed of light – even electromagnetic radiation can’t escape – this is a “black hole”

Atomic spectra and the Bohr theory of hydrogen Emission spectra of gases led Bohr to the conclusion that electrons exist in discrete orbits around nuclei Demo Hydrogen Helium Xenon

Atomic spectra and the Bohr theory of hydrogen The spectra exhibit narrow wavelengths of emission, which follow an empirical rule (for hydrogen) where R H is a constant (Rydberg constant) This dependence, and the discrete nature of the lines, cannot be explained with classical physics

Atomic spectra and the Bohr theory of hydrogen Bohr postulated a model that accounts for the observed phenomena. It’s assumptions are: 1.Electrons in circular orbits 2.Only certain orbits are stable 3.Electrons “jumping” between states conserves energy by absorbing or emitting radiation, with frequency proportional to the energy difference of the levels 4.The angular momentum of the orbits is quantized h=Planck’s constant =6.63  Js

Atomic spectra and the Bohr theory of hydrogen Electric potential energy due to positive charge of the nucleus In the same way as we calculated the total energy for planetary motion, we find

Atomic spectra and the Bohr theory of hydrogen Now solving the the expression for the quantization of the angular momentum for v, and setting these two equations equal yields From Newton’s 2 nd law n=1 defines the “Bohr radius”

Atomic spectra and the Bohr theory of hydrogen Using this to calculate energy differences between levels and the frequency of light corresponding to that energy, we find a general formula Quantized radii imply quantized energies: