Astro 110-01 Lecture 12 Energy and Gravity (Cont d) 13/02/09 Habbal Astro110-01 Lecture 12 1
Energy due to movement of Kinetic Energy: object E k = ½ m v 2 13/02/09 Habbal Astro110-01 Lecture 12 2
Gravitational Potential Energy E g = m g h On Earth, it depends on an object s mass (m). the strength of gravity (g). the distance an object could potentially fall. 13/02/09 Habbal Astro110-01 Lecture 12 3
Impact Velocity of a dropped Egg Orbital trajectory of cannonballs Height of a Ball Thrown into Air 13/02/09 Habbal Astro110-01 Lecture 12 4
Mass-Energy Mass itself is a form of potential energy. E = 2 mc A small amount of mass can release a great deal of energy. Concentrated energy can spontaneously turn into particles (for example, in particle accelerators). 13/02/09 Habbal Astro110-01 Lecture 12 5
Conversion of mass to energy Conversion of Mass to Energy 13/02/09 Habbal Astro110-01 Lecture 12 6
What have we learned? What keeps a planet rotating and orbiting the Sun? Conservation of angular momentum Where do objects get their energy? Conservation of energy: Energy cannot be created or destroyed but only transformed from one type to another. Energy comes in three basic types: kinetic, potential, radiative. 13/02/09 Habbal Astro110-01 Lecture 12 7
The Force of Gravity [Section 4.4 ] Our goals for learning: What determines the strength of gravity? How does Newton s law of gravity extend Kepler s laws? How do gravity and energy together allow us to understand orbits? How does gravity cause tides? 13/02/09 Habbal Astro110-01 Lecture 12 8
What determines the strength of gravity? The Universal Law of Gravitation: 1. Every mass attracts every other mass. 2. Attraction is directly proportional to the product of their masses. 3. Attraction is inversely proportional to the square of the distance between their centers. 13/02/09 Habbal Astro110-01 Lecture 12 9
Inverse square law of gravity Inverse Square Law for Gravity 13/02/09 Habbal Astro110-01 Lecture 12 10
Exercise Compare the strength of gravity between Earth and Sun to that between Earth and Moon M(Sun) = 1.99 10 30 kg M(Moon) = 7.35 10 22 kg d(earth-sun) = 1.5x10 8 km d(earth-moon) = 3.84 10 5 km G = 6.67 10-11 m 3 /kg s 2 179 13/02/09 Habbal Astro110-01 Lecture 12 11
g = 9.8 m/s 2 http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html 13/02/09 Habbal Astro110-01 Lecture 12 12
How does Newton s law of gravity extend Kepler s laws? Kepler s first two laws apply to all orbiting objects, not just planets. Newton: Ellipses are not the only orbital paths. Orbits can be: - bound (ellipses) - unbound parabola hyperbola 13/02/09 Habbal Astro110-01 Lecture 12 13
Orbit of comets: Example: Sun grazing comets 13/02/09 Habbal Astro110-01 Lecture 12 14
Newton s version of Kepler s Third Law p2 = 4"2 G(M 1 +M 2 ) a 3 p = orbital period a = average orbital distance (between centers) (M 1 + M 2 ) = sum of object masses G = 6.67 10-11 m 3 /kg s 2 13/02/09 Habbal Astro110-01 Lecture 12 15
Exercise: Newton s version of Kepler s Third Law p2 = 4"2 G(M 1 +M 2 ) a 3 Check that units of G are correct G = 6.67 10-11 m 3 /kg s 2 13/02/09 Habbal Astro110-01 Lecture 12 16
Application of Newton s version of Kepler s Third Law Newton s version of Kepler s Third Law: If a small object orbits a larger one and you measure the orbiting object s orbital period AND average orbital distance THEN you can calculate the mass of the larger object. Example: Calculate the mass of Sun from Earth s orbital period (1 year) and average distance (1 AU). 1.99 1030 kg 13/02/09 Habbal Astro110-01 Lecture 12 17
How do gravity and energy together allow us to understand orbits? More gravitational energy since further away, less kinetic energy since v is smaller Total orbital energy gravitational + kinetic stays constant if there is no external force. Less gravitational energy since closer to Sun, more kinetic energy since v is larger Orbits cannot change spontaneously. 13/02/09 Habbal Astro110-01 Lecture 12 18
Changing an Orbit So what can make an object gain or lose orbital energy? Friction or atmospheric drag A gravitational encounter Comet loses orbital energy to Jupiter, changing its unbound orbit around the Sun 13/02/09 Habbal Astro110-01 Lecture 12 19
Escape Velocity: The velocity needed for an object to completely escape the gravity of a large body such as moon, planet, or star If an object gains enough orbital energy, it may escape (change from a bound to unbound orbit). 13/02/09 Habbal Astro110-01 Lecture 12 20
Escape velocity For object to escape gravity must have: E k E p ½ m v 2 m g h Or for Earth, h = R v esc 2 g R 2 G M/R independent of mass of object 13/02/09 Habbal Astro110-01 Lecture 12 21
Escape velocity: Exercise Calculate escape velocity for Earth v esc 2 G M/R G = 6.67 10-11 M = 5.97 10 24 kg R = 6378 km V esc = 11 km/s 13/02/09 Habbal Astro110-01 Lecture 12 22
Escape velocity: Illustrated Escape velocity from Earth 11 km/s from sea level (about 40,000 km/hr) Escape velocity from Earth 13/02/09 Habbal Astro110-01 Lecture 12 23
Escape and orbital velocities don t depend on the mass of the cannonball. 13/02/09 Habbal Astro110-01 Lecture 12 24