Newton’s law of universal gravitation Liz Fox

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 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 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 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

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

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

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 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. The periodic times of the five primary planets are as the 3/2th power of their mean distances from the sun.

“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. 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.

An Inverse-Square Law Centripetal vs. centrifugal Centripetal vs. centrifugal Huygens- Horologium Oscillatorium (On Pendulum Clocks) Huygens- Horologium Oscillatorium (On Pendulum Clocks) When 2 identical bodies move with the same velocity on unequal circumferences, their [centripetal] forces are in the inverse proportion to their diameters 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 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

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. 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.

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. 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. It is solely the gravity of the earth that keeps the moon in orbit.

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. 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 ( ) Henry Cavendish ( ) Hypotheses non fingo Hypotheses non fingo

Gravitation for Extended Bodies Inside a homogeneous hollow spherical shell, a point mass experiences no net gravitational force 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 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 Same for solid sphere of uniform density Teachers' Domain: String Theory: Newton's Embarrassing Secret

Inertial and Gravitational Masses Inertial vs. Gravitational mass 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.” 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 3 rd Law Kepler’s 3 rd Law

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