#### Transcript Chapter 12

Chapter 12 - Gravity Newton’s Law of Gravitation Kepler’s Laws of Planetary Motion Newton’s -------- Force of Attraction - Weight G = 6.67 x 10-11Nm2kg-2 ME = 5.98 x 1024kg RE = 6.37 x 106 m m = 70 kg Calculate your Weight! Solution to Problem Cavendish Experiment Henry Cavendish – English Physicist Cavendish Experiment The Gravitational Field Radial Nature and Inverse Square law g = GME/R2 Calculate the field strength at any location The Acceleration Due to Gravity at a Height h Above the Earth’s Surface Orbital Motion Newton’s Cannon Newton visualized that if a projectile were fired with enough velocity the Earth would forever “curve” away and it would never hit the ground.. … It would be in orbit! Newton’s Cannon Weightlessness While falling you still have weight but your apparent weight is zero! Kepler’s Laws of Motion 1st Law 2nd Law Ellipses Equal Area 3rd Law R3/T2=constant Kepler's second law How fast does the planet move in its orbit? As the planet moves in its orbit, a line from the sun to the planet sweeps out equal areas in equal times. Is Pluto a Planet? Kepler's second law Kepler’s 3rd Law T2 / R3= constant Ta2 / Tb2 = Ra3 / Rb3 The ratio of the squares of the orbital periods for two planets is equal to the ratio of the cubes of orbital radius. Useful Links Kepler’s Laws Centripetal Forces on Satellites is supplied by the gravitational force. If the equations are set equal what do you get? Satellites and Orbital Velocity An expression for orbital velocity ! Quick Calculator Set the centripetal force equal to the gravitational force Vo2 = (GM/r) Orbital Velocity & Period Remember that for any object moving in a circular path, we can relate the speed and period? Vo = 2πR/T So,….. Satellites in circular orbits. Vo2 = (GM/r) Vo = 2πR/T If these equations are set equal what do you get? Kepler’s 3rd Law - Derived Square both sides and set equal. Vo 2= (2πR/T)2 Vo 2 = (GM/R) 4π2R2/T2 = GM/R Result… T2 / R3 = 4π2 / GM, K = 4π2 / GM Kepler’s 3rd Law T2 / R3= constant = (4π2/GM) T2 = (4π2/GM) R3 Where (4π2/GM) is the Kepler constant! Depends only on the Mass of the planet being orbited. Geosynchronous Orbit AKA: Geostationary Orbit or Parking orbit At what altitude must a satellite be placed in order to be in a geosynchronous orbit? Binary Star Systems m1a1 m2 a2 v 2 v 2 1 2 m1 m 2 r r 1 2 2r1 2 m1 T 2 r 2 2 m2 T r1 r2 m1r1 m2 r2 m2 r1 m1 r2 1 3 Both stars have the same orbital period. Gravitational Potential Energy Gravitational potential energy is energy an object possesses because of its position in a gravitational field. Gravitational Potential Energy between two masses increases as r increases. (becoming less neg) At ∞ it becomes 0 Joules m1m 2 U G r Gravitational Potential Energy Work done against the gravity force in bringing a mass in from infinity where the potential energy is assigned the value zero . m1m 2 U G r Gravitational Potential Energy Change in KE = Change in PE ΔKE + ΔPE = 0 If these equations are set equal what do you get? Escape Velocity ! Set Gravitational Potential Energy equal to Kinetic Energy. GMm/R = ½ mVe 2 Solve for V. V = √ (2GM/R) Gravitational Potential The potential energy that a unit mass (usually 1 kg) would have at any point. M is the mass of the gravitating object. This is sometimes useful because it assigns each point in space a definite gravitational potential value, irrespective of mass. Units are in Joules/kg V=- Tides Tides are due to differential gravitational forces on a body. Consider the Earth and Moon: the gravitational force on the Moon due to Earth is stronger on the near side than on the far side. This net difference in force will cause the body to stretch along the line between the bodies. Tides Tidal flexing and heating of Io causes the most active volcanism in the Solar System Tidal Friction Tides result in a net force which slows Earth’s rotation and speeds the Moon’s orbital velocity. As a result the day is getting longer by ~1 second/century and the distance between the Earth and Moon is increasing. There is evidence for this in the fossil record on Earth Tidal Effects Roche Limit: Tidal deformation increases as an astronomical object moves closer to the body it orbits. At the Roche Limit, the object can break into small pieces. Tidal Locking: occurs when one astronomical object always points its tidal bulge at the object it orbits. Roche limit: tidal forces destroy bodies which come too close to a source of gravitation through differential acceleration Consider an orbiting mass of fluid held together by gravity. Far from the Roche limit the mass is practically spherical. The body is deformed by tidal forces Within the Roche limit the mass' own gravity (self-gravity) can no longer withstand the tidal forces, and the body disintegrates. Particles closer to the primary orbit move more quickly than particles farther away, as represented by the red arrows. The varying orbital speed of the material eventually causes it to form a ring. http://www.answers.com/topic/roche-limit#wp-Fluid_satellites Tidal Locking What path does the moon trace during its orbit about the sun?