Gravitational Fields (AIS).ppt
- 1. Planetary Motion & Gravitational Fields
- 2. From • Close to earth the acceleration due to gravity changes slowly. • As the distance from earth increases, the change in the acceleration due to gravity increases. Variation of g 2 r GM g www.physicsclassroom.com/Class/energy/U5L1b.cfm
- 3. Variation of g On a line joining the centers of two point masses • If m1 > m2 then Note: when g = 0 the gravitational fields cancel each other out
- 4. Escape Velocity • The velocity needed to overcome (escape) the gravitational pull of a planet. • As the body is moving away from the planet, it is losing kinetic energy and gaining potential energy. • To completely escape from the gravitational attraction of the planet, the body must be given enough kinetic energy to take it to a position where its potential energy is zero.
- 5. Escape Velocity • The potential energy possessed by a body of mass m, in a gravitational field is given by • If the field is due to a planet of mass M and radius R, then the escape velocity can be calculated as follows: • So, • as g = GM/R² GPE = Φm ΔKE = ΔGPE E esc r GMm mv 2 2 1 R GM vesc 2 gR vesc 2
- 6. Satellites: Orbits & Energy using Newton’s second law (F=ma) where v2/r is the centripetal acceleration. Solving for v2 So, Total Mechanical Energy of a satellite is: PE = - GMm r r v m r GMm F 2 2 r GMm mv KE 2 2 2 1 KE = - PE 2 ME = KE+PE = GMm 2r - GMm r r GMm ME 2 r GM v 2
- 7. Satellites: Orbits & Energy This is for a circular orbit. • For a satellite in an elliptical orbit with a semi-major axis a where a was substituted for r a GMm E 2 ME = -KE
- 8. Energy Graphs • As shown the energies for a satellite are, • Graphing these, F = - GMm r r GMm KE 2 r GMm ME 2 http://www.opencourse.info/astronomy/introduction/06.motion_gravity_laws/energy_elliptical.gif
- 10. Energy Wells • If we rotate the potential energy graph, we can generate a 3D picture known as an energy well. http://www.opencourse.info/astronomy/introduction/06.motion_gravity_laws/
- 11. Energy Changes in a Gravitational Field • A mass placed in a gravitational field experiences a force. – If no other force acts, the total energy will remain constant but energy might be converted from g.p.e. to kinetic energy. • If the mass of the planet is M and the radius of the orbit of the satellite is r, then it can easily be shown that the speed of the satellite, v, is given by – if r decreases, v must increase. Physics for the IB Diploma 5th Edition (Tsokos) 2008
- 12. Energy Changes in a Gravitational Field • If the satellite’s mass is m, then the kinetic energy, K, of the satellite is • the potential energy of the satellite, U, is These equations show: – that if r decreases, K increases but U decreases (becomes a bigger negative number) – the decrease in U is greater than the increase in K. • Therefore, to fall from one orbit to a lower orbit, the total energy must decrease. – work must be done to decrease the energy of the satellite if it is to fall to a lower orbit. r GM m U r GM m K 2 1
- 13. Energy Changes in a Gravitational Field • The work done, W, is equal to the change in the total energy of the satellite, W = ΔK + ΔP. • This work results in a conversion of energy from gravitational potential energy to internal energy of the satellite (it makes it hot!). • Air resistance reduces the speed of the satellite along its orbit. This allows the satellite to fall towards the planet. As it falls, it gains speed. • So, if a viscous drag (air resistance) acts on a satellite, it will – decrease the radius of the orbit – increase the speed of the satellite in it’s new orbit.
- 14. Energy Changes in a Gravitational Field • In principle, the satellite could settle in a lower, faster orbit. • In practice it will usually be falling to a region where the drag is greater. It will therefore continue to move towards the planet in a spiral path.
- 15. Physics for the IB Diploma 5th Edition (Tsokos) 2008 Energy Changes in a Gravitational Field • Looking at this with a fbd of the satellite spiraling downward due to air resistance. – Note that the velocity is no longer perpendicular to the weight. – Due to this there is a component of gravity which is now acting in the direction of the velocity providing a net force which is accelerating the satellite. Physics for the IB Diploma 5th Edition (Tsokos) 2008
- 16. Weightlessness Astronauts on the orbiting space station are weightless because... a. there is no gravity in space and they do not weigh anything. b. space is a vacuum and there is no gravity in a vacuum. c. space is a vacuum and there is no air resistance in a vacuum. d. the astronauts are far from Earth's surface at a location where gravitation has a minimal affect. e. None of the above Source: http://www.physicsclassroom.com/Class/circles/u6l4d.cfm
- 17. Weightlessness Astronauts on the orbiting space station are weightless because... The astronaut and the space station are both free-falling together - Apparent Weightlessness • The gravitational force on the astronaut provided the needed centripetal acceleration for the astronaut to stay in orbit. • The space station remains in orbit because of the gravitational force on it. • However, since there is no contact force between the satellite and the astronaut here is an apparent weightlessness Source: Kirk, Tim, Physics for the IB Diploma, Oxford University Press 2007 Source: Kirk, Tim, Physics for the IB Diploma, Oxford University Press 2007 Weightlessness
- 18. Weightlessness Astronauts on the orbiting space station are weightless because...with calculations • Summing the forces on the astronaut • For circular motion • So, or • Recall for a satellite, • Substituting, • With no reaction force (normal) the astronaut feels weightless net N g F F F F r v m Fnet 2 (Will learn in the next unit) r v m F r Mm G F N 2 2 2 2 2 v r M G r m r v m r Mm G N r GM v 2 0 r M G r M G r m N Source: Physics for the IB Diploma 5th Edition (Tsokos) 2008