Classical Mechanics PHYS 2006 Tim Freegarde 2020 21

Classical Mechanics PHYS 2006 Tim Freegarde 2020 -21 15 Gravitation

Classical Mechanics LINEAR MOTION OF SYSTEMS OF PARTICLES centre of mass Newton’s 2 nd law for bodies (internal forces cancel) rocket motion rotations and infinitessimal rotations ANGULAR MOTION angular velocity vector, angular momentum, torque parallel and perpendicular axis theorems rigid body rotation, moment of inertia, precession conservative forces, law of universal gravitation GRAVITATION & KEPLER’S LAWS 2 -body problem, reduced mass planetary orbits, Kepler’s laws energy, effective potential NON-INERTIAL REFERENCE FRAMES NORMAL MODES centrifugal and Coriolis terms Foucault’s pendulum, weather patterns coupled oscillators, normal modes boundary conditions, Eigenfrequencies 2

Newton’s law of Universal Gravitation • Exact analogy of Coulomb electrostatic interaction • gravitational force between two masses and • gravitational field • gravitational potential • gravity is a CENTRAL FORCE (acts along line joining masses) • spherically-symmetric central forces are CONSERVATIVE 3

Gravitational PE near the Earth’s surface 4

Inverse-square law forces Coulomb Gravity charge mass coupling potential energy per unit charge unit mass potential electric field force per unit charge unit mass gravitational field charge density mass density Gauss’s law 5

Gravity anomaly • local depends upon • altitude • latitude (spheroid, centrifugal force) • local material density • tides • phase of Moon • gravity map typically shows Bouguer anomaly • difference from value expected for uniform smooth ellipsoid • corrected for altitude • 6

Gravity anomaly NASA/JPL/University of Texas Center for Space Research • gravity anomaly = variation from uniform solid • 7

Schiehallion – estimate of mass of the Earth Walk. Highlands • Nevil Maskelyne (Astronomer Royal) 1774 • 11. 6” difference between north & south • concluded Earth density 4, 500 kg m-3 (20% low) Wikipedia 8

Gravimetry Wikipedia Muquans • quartz spring balance • atom matterwave interferometry • 5 μGal • 1 μGal 9

Newton’s law of Universal Gravitation • Exact analogy of Coulomb electrostatic interaction • gravitational force between two masses and • gravitational field VECTOR • gravitational potential SCALAR • scalar addition may be simpler than vector addition 10

Gravitational attraction of a spherical shell 11

Gravitational attraction of a spherical shell 12

Gravitational attraction of a spherical shell 13

Gravitational attraction of a spherical shell • Exact analogy of Coulomb interaction • gravitational force between two masses • gravitational field • gravitational potential 14

Galilean equivalence principle • gravitational field • gravitational motion of mass inertial mass gravitational mass • equivalence principle Apollo 15, David R Scott (7 August 1971) www. youtube. com/watch? v=MJy. UDpm 9 Kvk history. nasa. gov/alsj/a 15. clsout 3. html nssdc. gsfc. nasa. gov/planetary/lunar/apollo_15_feather_drop. html the trajectory of a point-like mass in a gravitational field is independent of the composition and structure of the mass inertial mass = gravitational mass 15

Newton’s law of Universal Gravitation • Exact analogy of Coulomb electrostatic interaction • gravitational force between two masses and • gravitational field • gravitational potential • a spherically-symmetric mass distribution will have the same external field as a point mass of equal magnitude at the sphere’s centre • inertial mass = gravitational mass 16

Classical Mechanics PHYS 2006 Tim Freegarde