Lecture 16 - Gravitation and Light (2/19/96)

Seeds: Chapters 4, 5

1. Review of Gravitation
   • Newton's Three Laws of Motion: Inertia, F=ma, and F_12 = -F_21 (reaction).
   • Universal Gravitation: F = G M m / r^2
   • Centripetal acceleration: a = v^2/ r
   • Gravitation is universal: a falling apple is subject to the same laws (and forces) as an orbiting moon or planet! It was being able to calculate the accelerations of the Moon and planets that convinced Newton of the correctness of his theory, not just an apple falling on his head.
   • Kepler's 3rd Law: P^2 = a^3 follows from Newtons gravitational force plus centripetal acceleration to calculate velocity. Newton's modification to Kepler's law (M + m)P^2 = a^3 takes into account the masses of both the Sun and planet (or planet and moon). Note that since it derives from Newton's theory, Kepler's laws are also universal - they apply not just to the Sun and planets in our solar system, but to any orbiting bodies, including satellites.
   • Center of Mass (or barycenter): Because of Newton's 3rd Law of equal action and reaction, we know that if the Sun causes the Earth to orbit, then the Earth causes the Sun to orbit. We can find that for two isolated bodies in mutual orbit: m_1 r_1 = m_2 r_2 where r_1 and r_2 are the distances from the barycenter, or center of mass, of the system. The two bodies orbit mutually in similar circular or elliptical orbits such that they remain on a line through the barycenter with respective distances given by the above equation and on opposite sides.
   • Thus, the relative orbit of the two can be thought of as circular or an ellipse centered on one of the bodies with a semi-major axis a = r_1 + r_2. This is the familiar Keplerian orbit.
   • Weight vs. Mass: We are used to using the terms weight and mass interchangeably. In fact, weight refers to the gravitational force on a body while mass in a measure of the innate matter content of a body. The mass of an astronaut is the same on the Moon as on the Earth, but the weight is less on the Moon, because the gravitational force exerted by the Moon at the surface is less. In space, an astronaut is "weightless", not massless!
   • We can calculate the weight of a body at the surface of the Earth by adding up the contributions from each individual atom of the Earth. If we approximate the Earth as a sphere (of radius R) with us at the surface, and break up the Earth's mass into tiny cubes, and add up the forces, then take the limit as the little chunks become infintesimally small, then we find Newton's result that the gravitational acceleration at the surface is a = GM/R^2 where M is the mass of the Earth. Note that this is the same as if the entire mass of the Earth were concentrated at the center! This us a property of a sphere and the force falling as 1/R^2.
   • Thus, if we write the weight (force) at the surface as F = mg, then the acceleration of gravity g at the Earth's surface is given by g = GM/R^2 = 9.8 m/s^2.
   • Although we have done this only for illustrational reasons, I have shown you how to calculate the centripetal acceleration in an orbit and the gravitational acceleration at the surface of a sphere by using "infintesimals" - this is calculus as it was devised by Newton! We calculated the centripetal acceleration a = v^2/r by finding the small changes in x and y needed to stay on a circle. The calculus of changes involve derivatives or differential calculus. The adding up of tiny mass elements in the sphere is an example of integration or integral calculus.
2. Light and Electromagnetic Waves
   • Almost everything we know about the Universe has been brought to us by the light that we observe with our instruments and eyes.
   • Light, radio waves, radar, X-rays and ultraviolet rays are all examples of the same phenomenon of electromagnetic radiation.
   • Light can be thought of as a electromagnetic wave with crests and troughs as in ocean waves (but without the ocean!). The things that are rising and falling in an electromagnetic wave are the electric and magnetic "fields", which rise and fall in step as a function of space and time.
   • If we were to freeze time, and move along the direction of a wave, we would measure electric and magnetic forces that increased then decreased then changed direction and back again, like a sine wave. The distance between crests of the wave is called the wavelength and is measured in meters (or centimeters or nanometers, whatever is convenient).
   • If one were to let time move normally, but measure the electric and magnetic forces as a wave passes by, then we would see the crests and troughs moving past us at the speed of light. Thus, we would see the waves pass us at the frequency of a certain number of cycles per second, say f, given by: f = c / L where L is meant to be the wavelength. We are familiar in using frequency to describe radio waves. When we say our favorite radio station has a frequency of 100 Megacycles (or MegaHertz), we mean that f = 100 x 10^6, that is 10^8, cycles per second. (Remember, mega means 10^6). Note that since the speed of light c = 3 x 10^8 m/s, that a frequency of 100 megacycles corresponds to a wavelength L = c/f = 3 meters.
   • The electric and magnetic forces in an electromagnetic wave are at right angles (perpendicular) to each other always.
3. Light and Photons
   • Light can also be thought of as made up of particles, known as photons. Light is both a wave and a particle at the same time. The reconciliation of this dual nature has been one of the greatest breakthroughs in physics at the turn of the century.
   • Each photon has a specific energy given proportional to the frequency of the wave that it corresponds to. The energy of a photon of frequency f is: E = h f where the constant h is known as Planck's constant.
   • Planck's constant h = 6.63 x 10^-34 Joules sec, where the Joule (J) is the SI unit of energy (1 Joule = 1 kg m^2/ s^2), and is the energy needed to run a 1 Watt light bulb for 1 second (1 Watt = 1 Joule/sec) - or a 100 Watt light bulb for 0.01 seconds.
   • There are many different energy photons, with energies ranging from zero to infinity, each corresponding to waves with wavelengths from infintely long to infintesimally small (zero). This is what is called the electromagnetic spectrum.
   • Radio waves, with wavelengths around 1 meter, are fundamentally the same as visible light, with wavelength around 500 nanometers (nm), the same as X-rays, with wavelengths around 1 nm.
   • Because the photon energy is proportional to frequency, the high frequency, short wavelength radiation like UV and X-rays and Gamma Rays can actually do major damage to our bodies if we are exposed to it. It takes many microwaves (wavelength about 1 cm) to cook food in a microwave oven, because the photons each have very low energy. Nuclear explosions release large amounts of deadly Gamma Rays, each of which can do lots of molecular damage when they pass into organic material.
   • Different materials absorb specific frequencies of light. This is due to resonances with atomic and molecular structures as we will see later on. The air in the Earth's atmosphere absorbs the waves except special "windows" or wavelength bands where it is transparent. Both visible light and radio waves fall in these windows, so we can see out from the Earth into the heavens.
   • Just as electromagnetic waves of light are also particles (photons), all particles such as electrons and protons are waves! It turns out that a particle of momentum p has a wavelength of L = h / p = h / m v so more massive particles have smaller wavelenghths. This property is used in electron microscopes, which act like normal microscopes except using the wave nature of electrons instead of light. Because the wavelength of an electron wave is very very small, it can see smaller things than normal light based microscopes.

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