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Gravitation

Gravitation . Ch 5: Thornton & Marion. Introduction. Newton, 1666 Published in Principia, 1687 (needed to develop calculus to prove his assumptions) Newton’s law of universal gravitation

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Gravitation

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  1. Gravitation Ch 5: Thornton & Marion

  2. Introduction • Newton, 1666 • Published in Principia, 1687 (needed to develop calculus to prove his assumptions) • Newton’s law of universal gravitation • Each mass particle attracts every other particle in the universe with a force that varies directly as the product of the two masses and inversely as the square of the distance between them.

  3. Cavendish Experiment • Henry Cavendish (1731-1810) verified law and measured G • G=6.67 x 10-11 N m2 / kg2 • video

  4. Extended Objects

  5. Gravitational Field • Gravitational field = force per unit mass • For point masses: • For extended objects:

  6. White Boards • Is gravity a conservative forces?

  7. White Boards • Is gravity a conservative forces?

  8. Gravitational Potential • Gravitational field vector can be written as the gradient of a scalar function: • Φ is the gravitational potential • Energy/mass • We can obtain Φ by integrating:

  9. Potential from Continuous Mass Distributions Prime denotes integration element

  10. Gravitational Potential • Once we know Φ, we can determine the gravitational force and the gravitational potential energy.

  11. Example • What is the gravitational potential both inside and outside a spherical shell of inner radius b and outer radius a?

  12. Example • Astronomical measurements indicate that the orbital speed of masses in many spiral galaxies rotating about their centers is approximately constant as a function of distance from the center of the galaxy. Show that this experimental result is inconsistent with the galaxy having its mass concentrated near the center of the galaxy and can be explained if the mass of the galaxy increases with distance R.

  13. Poisson’s Equation • Gauss’s Law for the electric field • Gauss’s Law for gravity • Poisson’s Equation

  14. Lines of Force & Equipotential Surfaces • Equipotential lines connect points of constant potential • Force is always perpendicular to the equipotential lines • Like a contour map, lines of equipotential show where an object can move while maintaining constant gravitational potential energy

  15. Using Potential • Potential is a convenient way to calculate the force • Force is physically meaningful • In some cases, it might be easier to calculate the force directly • Potential is a scalar

  16. Example • Consider a thin uniform disk of mass M and radius a. Find the force on a mass m located along the axis of the disk. Solve this using both force and potential.

  17. Lagrange Points • Solved by Euler & Lagrange • Sun is M1 • Earth-Moon is M2 • Stable equilibrium • L4 , L5 • WMAP satellite in L2

  18. Ocean Tides • The Moon and Sun exert tidal forces on the Earth. This is because the strength of the gravitational force varies with distance, so that the near side of the Earth feels a larger force or acceleration than the far side. • We can differentiate the gravitational force equation to see how its strength varies over a distance dR.

  19. Tides • Continuing: Multiplying both sides bydR yields • If we want to figure out differential force across the size of the Earth, setdR= REarth . Then let d be the separation between M and m.

  20. Tides • Spring Tides occur when tidal forces from Sun and Moon are parallel. • Neap Tides occur when tidal forces from Sun and Moon are perpendicular. • Moon returns to upper transit 53 minutes later each day, so high tide occurs approximately 53 minutes later each day.

  21. White Boards • In the early 1980's the planets were all located on the same side of the Sun, with a maximum angular separation of roughly 90 degrees as seen from the Sun. This rough alignment was sufficient to make possible the Voyager spacecraft grand tour. Some people claimed that this planetary alignment would produce destructive earthquakes, triggered by the cumulative tidal effects of all the planets. Very few scientists took this seriously! To understand why, compute the max tidal effects on Earth produced by Jupiter (the most massive planet) and Venus (the closest planet). Compare these tidal effects to those caused by the Moon each month.

  22. Solution • Compute ratios of tidal forces from Jupiter and the Moon, and Venus and the Moon.

  23. MATLAB Problem • Start with the following code. Adjust the mass ratios and contour levels until you recreate the plot showing the Lagrange points. Name your file equipotential.m

  24. Elegant Universe • Gravity- From Newton to Einstein

  25. Rotation Curves of Galaxies

  26. An example • Determine the radial profile of the enclosed mass and the total mass within 8’.

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