Universal Gravitation Newton’s 4 th law

Universal Gravitation Kepler’s Laws Newton’s Law of Universal Gravity Applying Newton’s Law of Universal Gravity

Universal Gravitation Key Terms Tyconic system Kepler’s Laws Universal Gravity Kepler Constant Angular Momentum Eccentricity of Orbits Torsion Balance Microgravity Geostationary Orbit Perturbing Orbits

Universal Gravitation The Tyconic system was named after Tyco Brahe, a Danish astronomer who was a mentor to Johannes Kepler. In his system, the Earth was at the centre of the universe. The sun and moon orbited the earth and the other planets revolved about the sun.

Universal Gravitation lpc1.clpccd.cc.ca.us/.../ahistlec.htm

Kepler’s Laws Planets move in elliptical orbits, with the Sun at one focus of the ellipse. An imaginary line between the Sun and a planet sweeps out equal areas in equal time areas. The quotient of the mean radius of revolution cubed and the square of the period of revolution is constant and the same for all planets : r 3 /T 2 = k

Universal Gravity Newton reasoned that the attractive force which made an apple fall to the ground was the same force between the sun and the Earth and the Earth and the Moon. He concluded that these were force pairs and that every object must attract every other object. He summarized:

Universal Gravity The force of gravity between any two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centres. F 1,2 = G (m 1 m 2 ) / (r 1,2 ) 2 G is the universal gravitational constant (determined by Cavendish) as 6.67 x Nm 2 /kg 2 It should not be confused with g = 9.81 m/s 2

Reconciling Kepler’s 3 rd with Newton’s 4 th Consider the sun and a planet F grav = G (m s m p ) / (r) 2 Sun’s gravitational force supplies centripetal force for planet’s orbit, so G (m s m p ) / (r) 2 = m p v 2 /r but v = 2πr/T G m s / (r) 2 = (2πr/T) 2 and rearranging r 3 /T 2 = G m s / (4π) 2 Although this example involves the sun and a planet, it applies to all orbital motion

Finding the mass of the sun without a scale r 3 /T 2 = G m s / (4π) 2, rearranging, m s = (4π 2 /G)(r 3 /T 2 ) = 1.97 x kg