Alpha Centauri System consisting of three stars Alpha

Alpha Centauri � System consisting of three stars: Alpha Centauri A, Alpha Centauri B, and Proxima Centauri � Alpha Centauri A and B (depicted at left) form a binary star system � Binary star system: two stars orbiting around their center of mass � Video animation recorded at a speed 1, 000 x faster than real time What force is responsible for the motion of Alpha Centauri A and Alpha Centauri B?

Universal Gravitation

Gravity on Earth �Neglecting air resistance, all objects near the surface of the Earth are in free-fall �Know the acceleration due to gravity on the earth’s surface is 9. 8 m/s 2

Objectives �Discuss the historical development of the law of universal gravitation. �Understand how Newton’s law of universal gravitation explains both the motion of falling objects and the orbits of satellites and planets. �Understand how the acceleration due to gravity acting upon a mass is affected by the location and mass of the other object in question. �Quantitatively apply Newton’s law of universal gravitation to solve problems.

Newton and the apple �Fiction: Newton was sitting under an apple tree. Upon being struck upon the head by an apple, Newton realized gravity. �Fact: Newton observed an apple falling to the ground while sitting in his garden. He then reasoned that the same force that pulls an apple toward the ground is the same as the force that holds celestial objects in orbit. �Gravitational force – a force of mutual attraction between masses separated by a certain distance.

Inverse Square Relationship �Newton knew that the Moon was 60 x farther from the center of Earth than it was from the Earth’s surface. �If the force decreased at an inverse square rate, the gravitational force at the surface of the Moon would be 1/602 times the gravitational force on Earth’s surface. �During Newton’s time, the period of the Moon ( 27 days) and the mass and radius of the Earth were known. Using these values, Newton was able to determine the acceleration of gravity on the Moon. �Newton’s value was not exactly correct since the known values were not known to great precision. Using values known today, Newton would have been correct.

Philosophiae Naturalis Principia Mathematica �Commonly referred to as the Principia �Published July 5, 1687 �Newton discusses: �the Laws of Motion �the Law of Universal Gravitation �the derivation of Kepler’s Laws �harmonic oscillation �Detailed the law of universal gravitation in the third volume of the book - De mundi systemate (on the system of the world) �Gravitas, Latin: “weight”

Universal Gravitation �G: universal gravitation constant G = 6. 67 x 10 -11 N m 2/ kg 2 �m: mass of an object �r: the distance between the center of mass of the two objects �Fg is an attractive force that always exists between two masses, regardless of: �the medium separating them �their size or composition

The Gravitational Force Between a Point Mass and a Sphere �A satellite in orbit around a planet can be considered as a point mass and a sphere. �Fg is the same as if all the mass of the sphere was concentrated at its center (the center of mass).

The Gravitational Force �The gravitational forces that any two masses exert on each other are always equal in magnitude and opposite in direction. �The gravitational forces are an example of an action-reaction pair.

Universal Gravitation Constant �G = 6. 67 x 10 -11 N m 2/ kg 2 �Since G is a very small number �gravity has the lowest relative strength of the four fundamental forces �force of gravity is negligible unless a very large mass involved

The Gravitational Force and Newton's Second Law of Motion, F = ma �Know that the gravitational forces acting on two masses are equal and opposite. �The resulting acceleration of each mass is not necessarily equal and opposite. �Consider the gravitational force that arises due to your interaction with the Earth using Newton’s Second Law of Motion, F = ma.

The Gravitational Force and Newton's Second Law of Motion, F = ma

The Gravitational Force �The most important of the fundamental forces at long distances because of it’s infinite range � Explains free-fall motion on Earth, planetary orbits, and large-scale order of galaxies. � Can analyze the orbits of celestial objects to determine it’s distance from other celestial objects (the Sun) � Allows researchers to detect the presence of matter that cannot be detected by telescopes – dark matter �Acts universally on all matter � Unlike the electromagnetic force, the gravitational force acts universally on all matter since it does not depend on a mass’ electric charge.

Gravitational Field �Gravity is a field force �Gravitational field strength g where g = Fg/m �Gravitational field is a vector with magnitude g pointing in the direction of Fg �Gravitational field strength equals free-fall acceleration The blue arrows correspond to the magnitude of the gravitational field vectors of Earth’s gravitational field at that point.

Gravitational Field Strength �The acceleration due to gravity decreases slowly with increasing height (altitude or distance between the center of mass of the Earth and the object in question)

Gravitational Field Strength �At distances comparable to or greater than the radius of the Earth, the acceleration due to gravity decreases at a faster rate.

Weight �An object’s inertial mass is the same regardless of the acceleration due to gravity. �“Weight” = mass x gravitational field strength �on earth, weight = mass x 9. 8 m/s 2 �dependent upon gravitational field strength therefore weight changes with location � �m: your mass; M: mass of planet; r: planet radius �your weight on the surface of any planet will depend upon the planet’s mass and radius

Summary �The gravitational force is a field force that always exists between two masses, regardless of the medium that separates them. �The same law of gravity applies everywhere in the universe. �The magnitude of the gravitational force between two masses is given by the formula: �Fg between two masses is an action-reaction pair; however, the resulting acceleration of each mass due to Fg will differ (if unequal masses).

Assignment �Holt, Rinehart, & Winston: Chapter 7 �Read pages 263 – 264 �Complete problems: 39, 40 �Worksheet �Complete problems 19, 25, 27, 29