Chapter 6 - Gravitation Newton’s Law of Gravitation (1687) Kepler’s Laws Implications of Newton’s Law of Gravitation Gravitation near the Earth’s surface Superposition of forces Justification for Kepler’s second and third law Gravitational Potential Energy

Newton’s law of (universal) gravitation

Cavendish Experiment (1798)

Vector Form of Newton’s Law of Gravitation by on

Superposition of Forces m1 m2 m0 m3

Problem 1 Three masses are each at a vertex of an isosceles right triangle as shown. Write an expression for the force on mass three due to the other two. m1 r r m3 m2

Gravity near the earth’s surface

Kepler’s Laws The Law of Orbits The Law of Areas The Law of Periods All Planets move in elliptical orbits, with the sun at one focus. The Law of Areas A line that connects a planet to the sun sweeps out equal areas in equal times. The Law of Periods The square of the period of any planet is proportional to the cube of the semi major axis of its orbit

Kepler’s 2nd Law The Law of Areas A line that connects a planet to the sun sweeps out equal areas in equal times.

Justification of Kepler’s third law (for circular orbits around the sun)

Problem 2 Verify Kepler’s third law for the earth revolving around the sun. distance from sun to earth = 1.496 x 1011 m mass of sun = 1.99 x 1030 kg. r Ms

Problem 3 What would be the height of a satellite with a period of one day mass of earth = 5.98 x 1024 kg. h Re r me

Gravitational potential energy again

Escape velocity