Variation of g (acceleration due to gravity) - cal

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Variation of g (acceleration due to gravity)
A22 KANE
Topic code:
The acceleration on an object close to the Earth, due to the Earth's
gravitational field; symbol g. Its value is taken as 9·806 65 m/s2 (standard
acceleration of gravity, internationally adopted value), but its actual value
varies between 9·76 and 9·83 over the Earth's surface because of geological
variations, and decreases with height above sea level.
Explanation
The total acceleration of a body is found by vector addition of the opposite of
the actual acceleration (in the sense of rate of change of velocity) and a vector
of 1 g downward for the ordinary gravity (or in space, the gravity there). For
example, being accelerated upward with an acceleration of 1 g doubles the
experienced weight. Conversely, weightlessness means an acceleration of 1 g
downward in an inertial reference frame.
The value of g defined above is an arbitrary midrange value on the Earth,
approximately equal to the sea level acceleration of free fall at a geodetic
latitude of about 45.5°; The standard acceleration of free fall is properly
written as gn (sometimes g0) to distinguish it from the local value of g that
varies with position.
The units of acceleration due to gravity, meters per second squared, are
interchangeable with newtons per kilogram.
Variations of Earth's gravity
Gravity varies by altitude, latitude and local variation.
On the earth's surface, the gravity will depend on the location at which it is
measured, smaller at lower latitudes, for two reasons.
The first is that the rotation of the Earth imposes an additional acceleration on
the body that opposes gravitational acceleration. This effect on its own would
result in a range of values of g from 9.789 m·s at the poles.
The second reason is the Earth's equatorial bulge (itself also caused by
centrifugal force), which causes objects at the equator to be farther from the
planet's centre than objects at the poles.
The combined result of these two effects is that g is 0.052 m/s2 more, hence
the force due to gravity of an object is 0.5 % more, at the poles than at the
equator.
Calculated value of g
Given the law of universal gravitation, g is merely a collection of factors in that
equation:
F = GMm/r2
To find the acceleration due to gravity at sea level you can plug in values of G
and the mass (in kilograms) and radius (in meters) of the Earth to obtain the
calculated value of g:
g = GM/r2
This agrees approximately with the measured value of g. The difference may
be attributed to several factors:First, Earth is not homogeneous. Second, the
Earth is not a perfect sphere. Third, the choice of a value for the radius of the
Earth.( This calculated value of g does not include the centrifugal force effects
that are found in practice due to the rotation of the Earth)
There are significant uncertainties in the values of G and of m1 as used in this
calculation. However, the value of g can be measured precisely and in fact,
Henry Cavendish performed the reverse calculation to estimate the mass of
the Earth.
Why g varies from place to place
The downward force of gravity is opposed by an outward centrifugal force due
to the planet's rotation, which is greater at the equator than at a higher
latitudes. (The centrifugal force is "fictitious" in the sense that the real force
caused by rotation is the centripetal force; however, it is a convenient fiction
for the sake of calculations.) By itself, this effect would result in a range of
values of g from 9.789 m/s2 at the equator to 9.823 m/s2 at the poles. This
discrepancy is further accentuated because of the Earth's equatorial bulge,
which causes objects at lower latitudes to be further from the planet's center
than objects nearer the poles and hence subject to a slightly weaker
gravitational pull.
Overall these two effects result in a variation of 0.052 m/s2 (0.171 ft/s2) in the
value of g, which leads to a variation in the weight of an object by about 0.5%
depending on whether it is weighed at the equator or at one of the poles.
Taking an average over the whole surface of the Earth, physicists have arrived
at a standard value for g of 9.80665 m/s2. On other planets and moons the
values of the acceleration due to gravity may be very different, resulting in
different weights for the same object on these various worlds.
Examples:
The acceleration due to gravity is:
g’ = GM/(R+h)2
dividing equation by g
g’/g = Gm/(R+h)2
/ GM/ R2
Finally, we get g’/g = (1+h/R) -2
Expanding by binomial theorem and neglecting higher powers of h/R
g’/g = (1-2 h/R)
This expression indicates that the acceleration due to gravity decreases with
althtude.
References:
AS/A2 Physics Book
Advanced PHYSICS through diagrams
www.physicshelpforum.com/physics-help/motion-forces/1025www.daviddarling.info/encyclopedia/A/accgrav.html
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