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Teaching Physics Innovatively 2015. Puzzling problems on gravity adi 1 Tam as Tasn 17. August, 2015. 1 BME, Institute of Mathematics Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion Contents


  1. Teaching Physics Innovatively 2015. Puzzling problems on gravity adi 1 Tam´ as Tasn´ 17. August, 2015. 1 BME, Institute of Mathematics

  2. Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion

  3. Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion

  4. Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.

  5. Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.

  6. Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.

  7. Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion

  8. Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): g mL ¨ α = − mgL ( α + β ) , α L = β R m ◮ Result: � � R L R L L →∞ T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).

  9. Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): ~ mg T ~ mL ¨ α = − mgL ( α + β ) , α L = β R m ~ F grav ~ mg ◮ Result: β R � � R L R L L →∞ Earth T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).

  10. Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): ~ mg T ~ mL ¨ α = − mgL ( α + β ) , α L = β R m ~ F grav ~ mg ◮ Result: β R � � R L R L L →∞ Earth T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).

  11. Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .

  12. Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms r r P A of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .

  13. Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms r r P A of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .

  14. Deviation angle of hyperbolic orbits v 0 Problem: A comet passes by the Sun. p Determine its angle of deviation α in terms of the initial speed v 0 and the im- α pact parameter p . Hint: ◮ Apply energy and angular momentum conservation for the perihelion P and the point at infinity ◮ Use the geometry of the hyperbola: � α � = b c 2 = a 2 + b 2 , tan a , p = a , PF = c − b 2

  15. Deviation angle of hyperbolic orbits b a Problem: A comet passes by the Sun. c Determine its angle of deviation α in α/2 terms of the initial speed v 0 and the im- P F α pact parameter p . Hint: ◮ Apply energy and angular momentum conservation for the perihelion P and the point at infinity ◮ Use the geometry of the hyperbola: � α � = b c 2 = a 2 + b 2 , tan a , p = a , PF = c − b 2

  16. Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? Earth (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!

  17. Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? F E (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!

  18. Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? E F (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!

  19. Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.

  20. Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.

  21. Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.

  22. Motions observed from a space station Problem: A space station is orbit- y ing the Earth on a circular trajec- x tory, facing always with the same side towards the Earth. A small ω y object is thrown out of the space space Earth station station with a small initial veloc- ity. How does the object move x R relative to the space station? Remarks: ◮ Solve the problem in rotating reference frame ◮ Approximate the magnitude and the direction of the force vectors separately

  23. Motions observed from a space station Problem: A space station is orbit- y ing the Earth on a circular trajec- x tory, facing always with the same side towards the Earth. A small ω y object is thrown out of the space space Earth station station with a small initial veloc- ity. How does the object move x R relative to the space station? Remarks: ◮ Solve the problem in rotating reference frame ◮ Approximate the magnitude and the direction of the force vectors separately

  24. Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion

  25. The problem Let A be a fixed point in space at a distance d from a fixed sun S of mass M . Particles of mass m are shot from A in different directions at constant speed v . Which points can be reached by the particles? v m M A S d The enveloping curve of the orbits is to be found.

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