Universal Gravitation and Kepler’s Laws Physics 12

Newton’s Universal Law of Gravitation and Kepler’s First Law Newton had shown in his original article De Motu and later in Principia that the inverse square nature of gravity would lead to elliptical not circular orbits

Newton’s Universal Law of Gravitation and Kepler’s Second Law Even though planets are moving in elliptical orbits, the concepts of circular motion mostly apply Determine the speed of the Earth using the following data and assume that centripetal force is equal to gravitational force: Sun’s Mass – 1.99x1030kg Earth’s Mass – 5.98x1024kg Distance Earth to Sun (aphelion) – 152,171,522 km Distance Earth to Sun (perihelion) – 147,166,462 km

Newton’s Universal Law of Gravitation and Kepler’s Second Law Earth’s speed: Aphelion – 2.95x104m/s Perihelion – 3.00x104m/s This would indicate that Kepler’s Second Law is also supported by Newton’s Universal Law of Gravitation

Newton’s Universal Law of Gravitation and Kepler’s Third Law Since Kepler’s Third Law is a ratio of the orbtial radius cubed to the orbital period squared, we should be able to apply Newton’s Universal Law of Gravitation to the planetary motion to determine the value of the constant Set the centripetal force equation equal to Newton’s Universal Law of Gravitation Replace the velocity expression using orbital radius and orbital period

Newton’s Universal Law of Gravitation and Kepler’s Third Law This is called Newton’s version of Kepler’s Third Law

Mass of the Sun and Planets Henry Cavendish developed a torsion balance to determine the value of the Universal Gravitation Constant (approximately 70 years after Newton’s death) Using his apparatus, he was able to determine a value of G = 6.75x10-11Nm2/kg2 which is within 1% of the currently accepted answer Using G, he was then able to determine the mass of the Sun and other planets

Practice Problems Page 586 Questions 9-14