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Force, Acceleration, Momentum Newton’s Law: F = m a = Change in Momentum Change in Time F = ma = G M m d2 In this picture, what mass is m ? What mass is M? Which of the two rocks has greater acceleration? Gravity and Tides ✤ Caused by the difference in strength of the moon’s gravity on the near and far sides of the earth. Fwater drop = mdrop a = G Mmoon mdrop d2moon Fdrop = G Mmoon mdrop d21 Fdrop = G Mmoon mdrop d22 d2 d1 As the earth rotates, we experience two high tides and two low tides each day Gravity and Tides ✤ The sun also causes a differential force across the earth ✤ Spring tide: effect from sun and moon reinforce ✤ Neap tide: sun and moon effects fight each other, smaller tides ✤ Total change = few feet Spring Tides Neap Tides Gravity and Tides ✤ Friction drags the water along with the earth’s rotation, leads the moon ✤ Moon’s gravity pulls the earth toward it, slows earth’s rotation ✤ Earth’s gravity tries to speed up the moon in its orbit, move it to smaller distances from the earth. - Remember for a circular orbit around mass M: v = so if v increases, d decreases earth speeds up moon -> <- moon slows earth’s rotation drag force from Earth’s rotation GM √ d Mass, Weight, Force “Felix Baumgartner, Daredevil” Red Bull “Stratos” A balloon to 24 miles above the Earth, then jump Mass, Weight, Force What happens to Felix’s velocity as he drops? Does the force he feels change as he falls? Mass, Weight, Force ✤Radius of the earth: ~4000 miles ✤Distance from the surface of the earth when Felix jumped: 24 miles ➡Distance from the center of the earth when Felix jumped: 4024 miles ✤40002 = 16 million ✤40242 = 16.2 million ➡1% in the denominator of the force equation. Tiny change in Force F = GMm d2 Mass, Weight, Force If there were a scale under Felix’s feet as he dropped, what would it read? If he sat on a scale in his capsule, what would it read? Balloon vs. Rocket Moves away from earth due to force (thrust) generated by the engine Balloon vs. Rocket Kinds of Energy Kinetic energy = energy of motion: m v2 2 Potential energy = stored energy Gravitational potential energy = Energy stored by working against a gravitational force - GM1M2 d Notice resemblance to the equation for GM1M2 gravitational force: F = d2 Kinds of Energy m v2 Kinetic energy = energy of motion: 2 - GM1M2 Gravitational potential energy = d Energy units: kg m2/s2 (Joules) Other kinds of energy: Chemical potential: energy stored in chemical bonds Thermal: kinetic energy due to random motions of particles Radiative: light Energy In an isolated system: Energy is conserved sum of all the energy is a constant value Energy can change form: gravitational potential → kinetic radiative → thermal or be exchanged between objects Energy In an isolated system: Energy is conserved sum of all the energy is a constant value Examples of closed, isolated systems (includes all interacting objects): earth and pendulum a planet or planets orbiting a star All interacting objects may be the entire universe! Conservation of Energy: Examples Total Energy = Ekinetic + Egrav potential = Constant A swinging pendulum: each swing reaches the same height At top of swing, v=0 as the ball turns around: Ekinetic m v2 = 2 =0 Egrav potential = Etotal, largest possible value At bottom of swing, v = largest value Ekinetic = m v2 largest value 2 Egrav potential = minimum, ball is at its smallest distance from earth Conservation of Energy: Examples Total Energy = Ekinetic + Egrav potential = Constant A ball falling off a table On table: v=0 Ekinetic m v2 = 2 =0 Egrav potential = Etotal, largest possible value Right before ball hits floor: v = largest value Ekinetic = m v2 is largest 2 Egrav potential = smallest, ball is at its smallest distance from earth After the ball hits the floor, energy transferred to the floor: making noise, vibrations, thermal, ... Conservation of Energy and Orbits Total Energy = Ekinetic + Egrav potential = Constant Kepler’s 2nd Law: Planets sweep out equal area in equal time At perihelion: smallest distance Egrav potential = Ekinetic - GM1M2 d m v2 = 2 is largest At aphelion: largest distance - GM1M2 = d Egrav potential is largest (least negative) Ekinetic m v2 = 2 is smallest is smallest (a big negative number) Total energy: Kinetic Energy + Grav. Potential Energy How doesConservation energy conservation help us Orbits understand orb of Energy and Total Energy Ekinetic +by Egrav Constant Orbits are =defined thepotential total = energy of the orbiting Total goes energy: Kinetic Energy + Energy = consta Orbits are defined total energy of Grav. theon object orbit.energy fasterby or the slower, depending r,Potential butintotal never chan Bound Orbit: closed ellipse, object stays with the thing it is orbiting – Bound: PE Orbits are definedKE by +the Kinetic Energy + Potential Energy <total 0< 0 energy of the orbiting object. ! m v2 $ goes faster or slower, depending r, but ! onGM m $total energy never changes (con Kinetic Energy # 2 & + #" – R &% < 0 KE + PE > 0 " % – Bound: KE + PE < 0 ! m v 2 $ ! GM m $ & # &<# total en 2 !m v $ 2 m% $ " R % ! "GM Potential #Energy <0 & & + #– " R v 2 % GM " 2 % < ! m v 2 $ ! GM m 2$ R & 2GM # &<# " 2 % " Rv 2 <% KE + PE < 0 R v 2 GM < 2GM 2 R v< total en R Total energy: Kinetic Energy + Grav. Potential Energy = constant Conservation of Energy and Orbits Energy Ekinetic Egrav potential = Constant Orbits areTotal defined by =the total+ energy of the orbiting object. Orbitsor areslower, defineddepending by the totalon energy the object orbit. changes (consta es faster r, butoftotal energyin never – Bound Orbit: closed ellipse, object stays with the thing it is orbiting Kinetic Energy Potential Bound: KE+ + PE < 0Energy < 0 Potential Energy total energ 2$ ! ! $ GM m Kinetic m v & <0 # & + #– KE + PE > 0 Energy " 2 % " R % ! m v 2 $ ! GM m $ & # &<# " 2 % " R % v 2 GM < 2 R 2GM 2 v < R 2GM v< R total energ KE + PE < 0 Total energy: Kinetic Energy + Grav. Potential Energy = constant Conservation of Energy and Orbits Energy Ekinetic Egrav potential = Constant Orbits areTotal defined by =the total+ energy of the orbiting object. Orbitsor areslower, defineddepending by the totalon energy the object orbit. changes (consta es faster r, butoftotal energyin never – Bound Orbit: closed ellipse, object stays with the thing it is orbiting Kinetic Energy Bound: KE+ +Potential PE < 0Energy < 0 Potential Energy total energ 2$ ! ! $ GM m Kinetic m v & <0 # & + #– KE + PE > 0 Energy " 2 % " R % ! m v 2 $ ! GM m $ & # &<# " 2 % " R % v 2 GM < 2 R 2GM 2 v < R 2GM v< R total energ KE + PE < 0 otal energy: 2$ ! Kinetic Energy + Grav. Potential Energy = constant total energ ! $ mv GM m &= 0 # & + #– " 2 % R % " bits areare defined bybythe total energy of the orbiting object. Orbits defined the total energy of the orbiting object. Escape Speed 2 ! m v $ ! GM m $ ter or or slow, depending on r, but total energy never changes (constant # & on #slower, & =depending faster r, but total energy never changes (constant!) " 2 % " R % Kinetic Energy + Potential Energy = 0 for an orbit that is just barely unbound: 2 v Bound: KE =+ GM PE t UN-bound: KE<+0PE = 0 KE PE 2 R 2 2 $v $ !GM $ GM m ! m !vm ! $ m 2 + 2GM # & # " &2 v+% =# – #" – R & =&% 0< 0 " RR % " 2 % 2$ !m ! $ v GM m 2 ! m #v $ &!2GM $ & <GM m # # " v 2= & = %# " R & % " 2 % " RR % 2 v GM 2 < ellipse, object v closed GM Bound Orbit: R =2 stays with the it is orbiting 2 thing R 2GM 2 <Potential Energy < 0 Kinetic Energyv +2GM R v2 = R2GM v< 2GMR v= R total energy > total energ total energ total energy < total energ From last time: Orbits and Circular Motion ✤ Combine: Acceleration required to keep an object in circular motion: G M with acceleration from Gravity = d2 v2 GM = d d2 v= GM √ d v = speed for an object in stable circular motion around mass M at distance d v2 d Energy and Orbits v= GM √ d speed for an object in stable circular motion around mass M at distance d escape speed for an object at 2GM distance d from mass M √ d For an orbit, larger d = smaller v vescape = For an object in orbit at d, increase v by √2 and it will leave its orbit and escape. d Energy and Orbits v= GM √ d speed for an object in stable circular motion around mass M at distance d escape speed for an object at 2GM distance d from mass M √ d For an orbit, larger d = smaller v vescape = For an object in orbit at d, increase v by √2 and it will leave its orbit and escape. What happens if v increases by less than √2 ? A Speeds up B Slows down d Energy and Orbits v= GM √ d speed for an object in stable circular motion around mass M at distance d escape speed for an object at 2GM distance d from mass M √ d For an orbit, larger d = smaller v vescape = For an object in orbit at d, increase v by √2 and it will leave its orbit and escape. What happens if v increases by less than √2 ? A Speeds up B Slows down d Energy and Orbits v= GM √ d vescape = speed for an object in stable circular motion around mass M at distance d 2GM √ d escape speed for an object at distance d from mass M iClicker question: You are building a rocket to send the next rover to Mars. If the cost of the rocket depends on its escape speed from the Earth, where should you build the rocket? A B Earth the International Space Station Energy and Orbits v= GM √ d vescape = speed for an object in stable circular motion around mass M at distance d 2GM √ d escape speed for an object at distance d from mass M iClicker question: You are building a rocket to send the next rover to Mars. If the cost of the rocket depends on its escape speed from the Earth, where should you build the rocket? A B Earth the International Space Station Angular Momentum Momentum: mass x velocity change in momentum requires force Angular momentum: mass x velocity x radius The angular momentum of an isolated system is conserved. Just like linear momentum. r Angular Momentum Momentum: mass x velocity change in momentum requires force Angular momentum: mass x velocity x radius The angular momentum of an isolated system is conserved. Just like linear momentum. iClicker question: If this is a closed system, what happens to the speed v as I make the string length r shorter? A Stays constant, velocity is conserved B Slower C Faster r Angular Momentum Momentum: mass x velocity change in momentum requires force Angular momentum: mass x velocity x radius The angular momentum of an isolated system is conserved. Just like linear momentum. iClicker question: If this is a closed system, what happens to the speed v as I make the string length r shorter? A Stays constant, velocity is conserved B Slower C Faster r Angular Momentum and Orbits Kepler’s 2nd Law: Planets sweep out equal area in equal time At perihelion: smallest distance v must be largest At aphelion: largest distance v must be smallest Angular Momentum and Orbits Kepler’s 2nd Law: Planets sweep out equal area in equal time At perihelion: smallest distance v must be largest At aphelion: largest distance v must be smallest We came to the same conclusion thinking about energy: Egrav potential = Ekinetic - GM1M2 d m v2 = 2 At perihelion, d is smallest so potential energy is smallest, kinetic energy must be largest -> velocity is largest Angular Momentum Galaxies and the solar system form when clouds of stuff, mostly hydrogen, collapse due to their own gravitational force The clouds are very large. As they collapse, they become smaller. Conserve angular momentum → Spin up, become flat, disk-like. Angular Momentum and Galaxies