PHYS 2006

Tim Freegarde

Classical Mechanics

15 Gravitation2020-21

2

Classical Mechanics

LINEAR MOTION OF

SYSTEMS OF PARTICLESNewton’s 2nd law for bodies (internal forces cancel)

rocket motion

ANGULAR MOTION

rotations and infinitessimal rotations

angular velocity vector, angular momentum, torque

centre of mass

parallel and perpendicular axis theorems

rigid body rotation, moment of inertia, precession

GRAVITATION &

KEPLER’S LAWS

conservative forces, law of universal gravitation

2-body problem, reduced mass

NON-INERTIAL

REFERENCE FRAMES

centrifugal and Coriolis terms

Foucault’s pendulum, weather patterns

NORMAL MODESboundary conditions, Eigenfrequencies

coupled oscillators, normal modes

planetary orbits, Kepler’s laws

energy, effective potential

3

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

4

Gravitational PE near the Earth’s surface

5

Inverse-square law forces

Coulomb Gravity

charge mass

coupling coupling

potential potential

electric field gravitational field

charge density mass density

Gauss’s law Gauss’s law

Gauss’s law Gauss’s law

potential energy per unit mass

potential energy per unit charge

force per unit mass

force per unit charge

6

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

7

Gravity anomaly

• gravity anomaly = variation from uniform solid

•

NASA/JPL/University of Texas Center for Space Research

8

Schiehallion – estimate of mass of the Earth

WalkHighlands

Wikipedia

• Nevil Maskelyne (Astronomer Royal) 1774

• 11.6” difference between north & south

• concluded Earth density 4,500 kg m-3 (20% low)

9

Gravimetry

MuquansWikipedia

• quartz spring balance

• 5 μGal

• atom matterwave interferometry

• 1 μGal

10

Newton’s law of Universal Gravitation

• Exact analogy of Coulomb electrostatic interaction

• gravitational force between two masses and

• gravitational field

• gravitational potential

SCALAR

VECTOR

• scalar addition may be simpler than vector addition

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Gravitational attraction of a spherical shell

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Gravitational attraction of a spherical shell

13

Gravitational attraction of a spherical shell

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Gravitational attraction of a spherical shell

• Exact analogy of Coulomb interaction

• gravitational field

• gravitational potential

• gravitational force between two masses

15

Galilean equivalence principle

inertial mass

• gravitational field

• gravitational motion of mass

Apollo 15, David R Scott (7 August 1971) www.youtube.com/watch?v=MJyUDpm9Kvk

history.nasa.gov/alsj/a15/a15.clsout3.htmlnssdc.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

gravitational mass

• equivalence principle

inertial mass = gravitational mass

16

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

PHYS 2006

Tim Freegarde

Classical Mechanics