Gravitation Chapter 7

Planetary Motion & Gravitation 7.1

Nicholas Copernicus Since the beginning of time it was believed that everything revolved around the Earth. (Much like the typical teenager.) But, Nicholas Copernicus, Polish, noticed that the movement of the stars and planets didn’t agree with the Earth as the center.

His results were published in 1543 when Copernicus was on his death bed. His book showed an easy understanding of the Earth and other planets revolving around the Sun and not Earth.

Tycho Brahe Observed the eclipse of the sun in 1560 as a boy and decided to become an astronomer. After he grew up he used huge instruments that he designed and built to observe the skies. He spent 20 years recording exact positions of the planets.

He concluded that the Sun and moon orbit the Earth and that all other planets orbit the Sun.

Kepler’s Laws (Johannes Kepler) 29 year old German, an assistant of Brahe. Brahe trained Kepler to use his equipment and how to record the observations. Upon Brahe’s death, Kepler inherited 30 years worth of observations.

After studying the data he was convinced that the distance and motion of the planets could be explained by using geometry and mathematics. Kepler believed that the Sun exerted a force on the planets and placed the Sun at the center of the system. Kepler discovered three laws that describe the motion of every planet and satellite.

Kepler’s First Law The paths of the planets are ellipses with the Sun at one focus.

Kepler’s Second Law Kepler found that the planets move faster when they are closer to the Sun and slower when they are further away from the Sun. Second Law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals.

Kepler’s Third Law He also found a relationship between the periods of the planets and the mean distances away from the Sun. Third law states that the square of the ratio of the periods of any two planets revolving about the Sun is equal to the cube of the ratio of their average distances from the Sun.

The first two laws apply to each planet, moon and satellite individually The third law relates to the motion of several objects about a single body Planets to the Sun or Moon and Earth or Earth and satellites, etc…

Newton’s Law of Universal Gravitation 1666 Newton began his studies of the planetary motion Results find that force between bodies varies inversely with the square of the distance between the centers of the planet and Sun

His own third law states that the force one object applies to the second object is equal to the force the second applies to the first. The gravitational force is the attraction between these two objects and is proportional to the objects’ masses. He states that this applies even if one object isn’t a planet.

He proposed his law of universal gravitation. Objects attract other objects with a force that is proportional to the square of the distance between them.

Universal Gravitation & Kepler’s Third Law Consider a planet orbiting the Sun (assume a circular orbit) Newton’s 2 nd law F net = ma So F net = m p a c, where F is the gravitational force, m p is the mass of the planet, a c is the centripetal acceleration of the planet or

and then set equal simplify

Measuring the Universal Gravitational Constant It took 100 years from the time of Newton’s work until scientists developed the apparatus to measure the constant Henry Cavendish, English, used the following equipment to measure G

Suspended by thin wire so it can rotate Two large spheres and two small spheres close together The attraction between the spheres causes them to rotate When the force required to twist the wire equaled the gravitational force between the spheres the rods stopped moving Substituting the values for masses and distances into Newton’s law of universal gravitation he experimentally found G G=6.67 x N. m 2 /kg 2

The Importance of G This experiment, sometimes called “weighing the Earth,” allowed for a way to determine the mass of the Earth. and so