Geology R1031 and R5061

Handout #2

Determining the Mass and Density of the Earth

Modified from The Evolving Earth, Sawkins and others, 1974, Macmillian, pp.89-90 (See also Skinner and Porter pp. 28-29)

Galileo Galilei (1564-1642) noted that the velocity (v = distance/time) of freely falling objects increased (accelerated) with time and was independent of mass:

acceleration = v / t


and that the acceleration of freely falling objects at the surface of the Earth was ~ 9.8 meters per second.

Isaac Newton (1643-1727) derived a formula for the displacement of mass:

F (force) = M (mass) × A (acceleration)                 (1)

and formulated the universal law of gravitation for masses separated by a distance r:

F = G × m1 × m2 / r2 ,                 (2)
where
F - gravitational force,
G - universal gravitational constant,
m1 - mass 1,
m2 - mass 2.

The constant G must be quite small, because gravitational forces become large only when the masses are very large. It is not difficult to measure the force (F) that the Earth exerts at its surface on a mass m, of 1 kilogram (from equation 1):

F = m × A = weight of one kilogram,                 (3)
where
m - one kilogram mass,
A - acceleration due to gravity at the Earth's surface.

Newton also showed that any body with spherically symmetrical mass could be considered, for gravitational purposes, to have all its mass concentrated at its center. This meant that the quantity r in equation (2) is just the radius of the Earth. Three of the five quantities in Newton's equation (2) applied to the gravity of the Earth were thus known in Newton's time. All that was needed to find the mass of the Earth, was the gravitational constant (G).

Robert Hooke (1635-1703) formulated the relationship between applied forces (stress) and deformation (strain). For the force required to twist a suspended fiber as shown opposite:

F × L = T = K × a ,                 (4)
where
F - Force,
L -distance,
T - Torque,
K - fiber constant,
a - angle of twist.

Around 1800, Henry Cavendish (1731-1810) devised an experiment based on the weak gravitational attraction of two large lead balls as shown opposite. Cavendish combined Newton's equation (2) and Hooke's equation (4):

K × a / L = F = G × m1 × m2 / r2                 (5)

Knowing K, L, and the masses and rearraning equation (5), Cavendish arrived at a value of G from measured values of a and r:

G = (K × a × r2) / (L × m1 × m2 )                 (6)

Rearranging Newton's equation (2), using equation (3) and his value for G, Cavendish was in a position to "weigh the Earth".
(The presently accepted value of G is 6.672×10 -11 Newton × m2 / kg2 where Newton = kg × m / sec2)

M = (m × A × r2 ) / ( m × G) ,                 (7)
where
M - mass of the Earth,
m - one gram mass,
A - acceleration due to gravity at the Earth's surface,
r - radius of the Earth.

M = (A × r2 ) / G,                 (8)
where
M - mass of the Earth,
m - one gram mass,
A - acceleration due to gravity at the Earth's surface.

In Exercise 1 you will use Eratosthenes' method (Handout #1) to estimate the radius of the Earth and equation (8) to calculate its mass.

URL: http://www.cs.umn.edu/~checheln/geo/handouts/2.html
Comments to: visl0001@tc.umn.edu
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