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(b) What would happen if the ocean-basin depth were to change to (i) 1 km, (ii) 3 km
or (iii) 8 km?
13. Calculate the depths and densities beneath a 5-km-high mountain chain in isostatic
equilibrium with a 35-km-thick continental crust of density 2.8
×
10
3
kg m
-3
and a
mantle of density 3.3
×
10
3
kg m
−
3
by using the hypotheses of (a) Pratt and (b) Airy.
14. A mountain range 4 km high is in isostatic equilibrium.
(a) During a period of erosion, a 2 km thickness of material is removed from the
mountains. When the new isostatic equilibrium is achieved, how high are the
mountains?
(b) How high would they be if 10 km of material were eroded away?
(c) How much material must be eroded to bring the mountains down to sea level?
(Use crustal and mantle densities of 2.8
×
10
3
kg m
−
3
and 3.3
×
10
3
kg m
−
3
,
respectively.)
15. A 500-m-deep depression on the Earth's surface fills with (a) sandstone of density
2.2
×
10
3
kg m
−
3
and (b) ironstone of density 3.4
×
10
3
kg m
−
3
. Assuming that
isostatic equilibrium is attained, calculate the thicknesses of sediment that will be
deposited in these two cases. (Use crustal and mantle densities of 2.8
×
10
3
and
3.3
×
10
3
kg m
−
3
, respectively.)
16. If subduction doubled the thickness of the continental crust, calculate the elevation
of the resulting plateau. Assuming that all plateaux are eroded down to sea level,
calculate the total thickness of material that would be eroded.
17. Assume that the oceanic regions are in Airy-type isostatic equilibrium. If the litho-
sphere has a uniform density, show that the depth of the seabed is given by
d
+
(
L
(
t
)
−
L
(0))
ρ
1
−
ρ
a
ρ
a
−
ρ
w
where
L
(
t
)isthe thickness of the lithosphere of age
t
,
d
is the depth of water at the
ridge axis and
ρ
w
,
ρ
1
and
ρ
a
are the densities of water, lithosphere and asthenosphere,
respectively.
18. (a) Calculate the maximum gravity anomaly due to a sphere of radius 1 km with
density contrast 300 kg m
−
3
buried at a depth of (i) 1 km, (ii) 2 km and (iii) 15 km.
(b) Calculate the gravity anomalies at distances of 1, 5 and 10 km from these spheres.
19. (a) What is the maximum gravity anomaly due to a cylinder of radius 1 km with
density contrast 200 kg m
−
3
buried at a depth of 1 km.
(b) Acylinder, of radius 50 km and buried at a depth of 100 km, yields the same
maximum gravity anomaly as that in (a). Calculate the density contrast of this
deep cylinder.
(c) Can gravity measurements resolve deep anomalous mantle density structures?
20. (a) Calculate the geoid height anomaly due to a mountain of height
h
for Pratt-type
compensation. What is this anomaly for
h
=
2 km?
(b) Calculate the same anomaly for an ocean depth
d
. What is this anomaly for
d
=
5 km?
21. What is the anomaly in the gravitational potential as measured on the spheroid if the
geoid height anomaly is (a) 3 m, (b)
−
5m and (c) 8 m?
