Lecture 10 Gravitational force (chap 7, 9.6)

Gravitation Gravitation is a fascinating phenomena. It was studied by Newton and Einstein, two of the greatest physicists in history However, compare to other three forces, gravity is the least known still. There are many mysteries about gravity we have not been able to solve. We need another Newton or Einstein!

Motion of planet Copernicus (1473-1543) published in 1543 his idea that the Earth is moving around the sun. Tycho Brahe (1546-1601) measured many data on the planet motion to prove or disapprove his theory. Johannes Kepler (1571-1630) summarized Brahe’s data into three laws. Kepler lived in Regensburg, Germany, outside of Munich in his last years.

Kepler’s three laws Each planet moves around the sun in an ellipse, with the sun at one focus. The radius vector from the sun to the planet sweeps out equal areas in equal interval of time. The squares of the periods of any two planets are proportional to the cubes of the semi-major axes of their respective orbits: 𝑇 ~ 𝑎 3/2 Kepler’s laws were initially ignored (Galileo and Descartes), only get very popular after his death.

Central force: Hooke What is the dynamical origin of the Kepler’s laws? The force must be a central force according to Galileo’s principle of inertia. Hooke might be the first one to realize this! "I will explain," says Hooke, in a communication to the Royal Society in 1666, "a system of the world very different from any yet received. It is founded on the following positions. 1. That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but that they also mutually attract each other within their spheres of action. 2. That all bodies having a simple motion, will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, an ellipse, or some other curve. 3. That this attraction is so much the greater as the bodies are nearer. As to the proportion in which those forces diminish by an increase of distance, I own I have not discovered it...

Newton’s inverse square law (1686) The central attractive force between any two objects can be precisely described by the following equation. 𝐹=𝐺 𝑚 𝑚 ′ 𝑟 2 where 𝑚 is the gravitational mass, or charge, which may or may not be the same as the inertia. 𝑟 is the distance between the object. 𝐺 is the gravitational constant. Using the vector notation (force of an object 2 on 1 𝐹 12 =𝐺 𝑚 1𝐺 𝑚 2𝐺 𝑟 12 3 ( 𝑟 2 − 𝑟 1 ) Where 𝑟 12 =| 𝑟 1 − 𝑟 2 |.

Solving newton’s equation

Universality The gravitational force is not just between the sun and planet. It applies to between moons and planets (Jupiter, earth) It applied between the object and the earth. He found that is the same as between the moon and the earth (read Feynman 7-4) . When extrapolated, Cavendish showed that this is true between any two objects on the earth.

Understanding the tides When the moon attracts the sea water closer, the latter will produce a high tide. However, the high tide does not happen once in every 24 hrs, when the location is closest to the moon. In fact, the high tide happens twice a day, every 12 hrs. because when the water is farthest from the moon, it feels less attraction than the average！

Application of the newton’s theory in solar system Orbits of the Jupiter and Saturn shall not be perfect ellipses. They wobble around the correct elliptical orbits. Uranus was a little weird. This lead to the prediction of another planet, Neptune, which was later discovered.

Gravitational attractions

Gravitational constant Gravitational constant was first measured by Cavendish through a torsion fiber. Cavendish, Henry (1798). "Experiments to Determine the Density of Earth".  Philosophical Transactions of the Royal Society of London 88: 469–526. when G is know, and the earth radius r and gravitational acceleration g are know, we can calculate the mass of the earth！ This was done 100 years after Newton’s work.

Inertial mass and gravitational mass. When the inertial mass is the same as the gravitational mass, any object will move exactly the same way in a gravitational field. This has been proved to high precision through expts like Galileo, Eotvos, Dicke. Einstein used this assumption to propose the general theory of relativity such that the gravitational interaction does not happen in a instant but only at the speed of light. This general theory has a lot of predictions.

Einstein theory of general relativity Time is not absolute. Different gravitational fields will have different rate of time or clock. Used in GPS. Anything with energy (not just mass) will have gravitational attraction. So when light passes through the sun, it will get distorted. Gravitational lensing! The interaction between masses is not just inverse square law. It has a small correction. This causes the extra procession of the mercury. Black-holes!

Not the final theory All other forces are consistent with quantum mechanics, but gravity is not. What is quantum gravity? Why is gravity so weak compared with other force? Is gravity modified at small distance , where space time may have extra dimension? What is the gravity at large distance? What is the dark energy and dark matter? Where is the gravitational wave?