The Physics
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Opus in profectus

# Universal Gravitation

## Summary

• Gravitational Force
• Newton's law of universal gravitation states that any two objects with mass will experience a gravitational force that is…
• universal (acts on all objects)
• attractive (there is no such thing as antigravity)
• directly proportional to the mass of each object (mass makes gravity)
• inversely proportional to the square of the separation between their centers (inverse square rule)
• Equations
magnitude notation
 Fg = − Gm1m2 r2
 Fg = magnitude of the gravitational force between two objects (Note: A minus sign is often added to the equation to show that the force is attractive. It is frequently ignored in practical calculations.) G = universal gravitational constant, often shortened to gravitational constant, or referred to informally as Big G (6.67 × 10−11 N m2/kg2) m1, m2 = mass of the two objects r = separation between the centers of the two objects
vector notation
 Fg = − Gm1m2 r̂ r2
 Fg = gravitational force vector between two objects r̂ = unit vector along the line separating the centers of the two objects (Note: The use of the minus sign here shows that the force vector points in the opposite direction of the separation vector. The force and separation have opposite sense.)
• Since the universal gravitational constant is a "small" number, gravity is a "weak" force.
• The only object that exerts an appreciable gravitational force on other objects in daily life is the Earth.
• We perceive the gravitational force of the Earth because it is massive (5.97 × 10−24 kg) and we are not normally far from it (usually less than 6,400 km from its center).
• A typical human can easily push up on a small patch of ground with a normal force that is equal to or greater than the gravitational force of the entire Earth pulling them down.
• The gravitational force between objects has infinite range.
• There is no value of the separation, r, that can make Fg = 0 in Newton's law of universal gravitation.
• Gravity can be reduced (by moving far away from massive objects), but it can never be eliminated.
• Gravity is the dominant force in the universe when energies are low and volumes are large (solar systems, galaxies, clusters, superclusters, and larger structures).
• Field
• A field is…
• a physical quantity that has a value (or set of values) at every point in space and time
• a mathematical function that returns a value (or set of values) for every point in space and time
• a way to deal with philosophical arguments against action at a distance
• Fields can be…
• scalar fields associated with quantities such as…
• temperature
• pressure
• density
• electric charge
• vector fields associated with…
• the flow of fluids (the velocity field) such as…
• wind
• ocean currents
• solar wind
• non-contact forces (force fields) such as…
• gravitational field
• electric field
• magnetic field
• tensor fields associated with…
• continuum mechanics quantities, namely the…
• stress tensor
• general relativity quantities such as the…
• metric tensor
• stress-energy tensor
• Ricci curvature tensor
• Gravitational Field
• Magnitude  g = Fg m
Where…  g = gravitational field (gravitational field strength) Fg = gravitational force on test mass m = mass of test mass
• Units  ⎡⎢⎣ N = m ⎤⎥⎦ kg s2
• Direction
• Gravity is described by a vector field.
• The direction of the gravitational field at any point in space is the direction of the net gravitational force on a "small" test mass.
• The gravitational field around a spherically symmetric mass is radial and points inward.
• Use vector addition to find the net gravitational field when more than one massive object is present.