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Newton's Law of Gravitation and Acceleration due to Gravity, Slides of Geology

Newton's law of gravitation and how it applies to the earth, allowing us to calculate the acceleration due to gravity. It also discusses the earth's mass estimation and the impact of earth's non-spherical symmetry on gravity measurements.

Typology: Slides

2012/2013

Uploaded on 07/19/2013

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Newton’s Law of Gravitation
For the gravitational attraction due to a hollow shell or
uniform sphere:
The force is the same as that of a point source of the same
mass located at the center of the sphere
This is true only outside the sphere.
At the center, the gravitational force must be zero (vectors all cancel
due to symmetry)
Because Earth is approximately
spherically symmetrical (recall this from
the seismology chapter?), we can treat
the Earth as a point source of mass
located at the center of the Earth!
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Newton’s Law of Gravitation

• For the gravitational attraction due to a hollow shell or

uniform sphere:

  • The force is the same as that of a point source of the same

mass located at the center of the sphere

  • This is true only outside the sphere.
    • At the center, the gravitational force must be zero (vectors all cancel

due to symmetry)

Because Earth is approximately spherically symmetrical (recall this from the seismology chapter?), we can treat the Earth as a point source of mass located at the center of the Earth!

Acceleration Due to Gravity

  • Because Earth acts like a point source of mass…
    • We can use Newton’s universal gravitation eq to calculate gravitational force on a small mass, m (^) s , on the surface of the Earth

To consider acceleration…

  • Insert Newton’s second law to solve for the force on the small body…
  • The small body's mass cancels out, and we are left with the equation for the acceleration due to gravity, g.
  • This relationship implies that all bodies on earth’s surface should experience the same falling acceleration

2 E

E s

R

M m

F = G

2 E

E s s

R

M m

F = m g = G

2 E

E

R

M

g = G

  • Although the mass of Earth is not practical to measure, it can

be easily calculated:

  • G = universal gravitational constant = 6.672 x 10 -8^ m^3 ⋅Mg-1⋅s -
  • g = 9.81 m⋅s -2^ (can be measured for falling objects)
  • RE = can be measured from surveys or geometric observations
    • Eratosthenes (~200 B.C.) ~ 6370 km
  • ME = 5.97 x 10 21 Mg or 5.97 x 10 24 kg
  • This mass estimate implies that the average density of the

Earth is ~ 5.5 Mg⋅m-

  • Most rocks are 2-3 Mg⋅m-
  • So, the interior of the Earth must be more dense than typical crustal rocks. Why? - Increasing lithostatic stress with depth squeezes rocks, increasing density - The core of the Earth is metallic (mostly iron ρ ≈ 7.8 Mg⋅m-3^ )

2 E

E

R

M

g = G

Densities of Common Rocks

• Rocks have a range of densities* (*your book gives more)

  • But in general, rock density does not widely vary
  • What does this say about typical gravity anomaly sizes?

Type Rock Density Unconsolidated Sand 1400-1650 kg/m^3 Sedimentary Salt 2100- Limestone 2000- Shale 2000- Igneous Granite 2500- Basalt 2700- Metamorphic Quartzite 2600- Gneiss 2600- Ore Galena 7400- Pyrite 4900- Magnetite 4900-