Geography Reference
In-Depth Information
Using the identity ∂p/∂z
=
(∂s/∂z)(∂p/∂s), we can express (1.27) in the
alternate form
∂p
∂x
∂p
∂x
z
+
∂z
∂x
∂p
∂s
∂s
∂z
s
=
(1.28)
s
In later chapters we will apply (1.27) or (1.28) and similar expressions for other
fields to transform the dynamical equations to several different vertical coordinate
systems.
PROBLEMS
1.1.
Neglecting the latitudinal variation in the radius of the earth, calculate the
angle between the gravitational force and gravity vectors at the surface of
the earth as a function of latitude. What is the maximum value of this angle?
1.2.
Calculate the altitude at which an artificial satellite orbiting in the equatorial
plane can be a synchronous satellite (i.e., can remain above the same spot
on the surface of the earth).
1.3.
An artificial satellite is placed into a natural synchronous orbit above the
equator and is attached to the earth below by a wire. A second satellite is
attached to the first by a wire of the same length and is placed in orbit directly
above the first at the same angular velocity. Assuming that the wires have
zero mass, calculate the tension in the wires per unit mass of satellite. Could
this tension be used to lift objects into orbit with no additional expenditure
of energy?
1.4.
A train is running smoothly along a curved track at the rate of 50 m s
−
1
.A
passenger standing on a set of scales observes that his weight is 10% greater
than when the train is at rest. The track is banked so that the force acting
on the passenger is normal to the floor of the train. What is the radius of
curvature of the track?
1.5.
If a baseball player throws a ball a horizontal distance of 100 m at 30
◦
latitude in 4 s, by how much is it deflected laterally as a result of the rotation
of the earth?
1.6.
Two balls 4 cm in diameter are placed 100 m apart on a frictionless horizontal
plane at 43
◦
N. If the balls are impulsively propelled directly at each other
with equal speeds, at what speed must they travel so that they just miss each
other?
1.7.
A locomotive of 2
10
5
kg mass travels 50 m s
−
1
along a straight horizon-
tal track at 43
◦
N. What lateral force is exerted on the rails? Compare the
magnitudes of the upward reaction force exerted by the rails for cases where
the locomotive is traveling eastward and westward, respectively.
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