Newton’s law of universal gravitation

Newton’s law of universal gravitation

• Liz Fox
• 2-16-06

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

Some background

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

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

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.

“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.

An Inverse-Square Law

• 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

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 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.

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

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

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

A Final Thought

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