Newton’s law of universal gravitation

1. Newton’s law of universal gravitation Liz Fox 2-16-06

2. A little review… • Law 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus. • Law 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time • Law 3: The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes http://www.edumedia.fr/m185_l2-newton-laws.html

3. Some background • Copernicus- De revolutionibus orbium coelestium – 1543 • Kepler- Astronimia Nova – 1609 • Galileo- Sedereus Nuncius - 1610

4. Newton’s Principia • Mathematical Principles of Natural Philosophy • Published in 1687 • Uses Kepler’s Laws to prove elliptical orbits • Explains behavior of tides, precession of the equinoxes, and the irregularities in the moon’s orbit

5. Newton’s Astronomical Data and Deductions • The planets orbiting Jupiter (Saturn) describe areas proportional to the times of descriptions; and their periodic times are as the 3/2th power of their distances from its center. • The periodic times of the five primary planets are as the 3/2th power of their mean distances from the sun.

6. “The nature of the forces” • The forces by which the primary planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the sun; and are inversely as the squares of the distances of the places of those planets from the sun’s center.

7. Centripetal vs. centrifugal Huygens- Horologium Oscillatorium (On Pendulum Clocks) - 1673 When 2 identical bodies move with the same velocity on unequal circumferences, their [centripetal] forces are in the inverse proportion to their diameters When identical bodies move on unequal circumferences with unequal velocities the [centripetal] force of the faster is to that of the slower as the square of their velocities An Inverse-Square Law

8. Newton’s take • The centripetal forces of bodies tend to the centers of the same circles; and are to each other as the squares of the arcs described in equal times divided respectively by the radii of the circles.

9. The Moon’s Centripetal Acceleration • The moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit. • It is solely the gravity of the earth that keeps the moon in orbit.

10. The Law of Gravitation for Point Masses • Law of universal gravitation- there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain. • Henry Cavendish (1731-1810) • Hypotheses non fingo

11. Gravitation for Extended Bodies • Inside a homogeneous hollow spherical shell, a point mass experiences no net gravitational force • Next, if a point mass is placed outside the shell, it is attracted to the exact center as if all of its mass were concentrated at a point • Same for solid sphere of uniform density Teachers' Domain: String Theory: Newton's Embarrassing Secret

12. Inertial and Gravitational Masses • Inertial vs. Gravitational mass • Inertial mass vs. weight – “The mass is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums.” • Kepler’s 3rd Law

13. A Final Thought • “Nature and nature’s laws lay hid in night; God said ‘Let Newton be!’ and all was light.”