This month's Challenge Question: Specs & Techs from GlobalSpec:
Two
straight tubes are drilled between two points on the earth, as shown in the
figure. A plastic object is dropped into each tube. Each object is a replica of
the other.
(a) Which object reaches the other end of the tube in the shortest
time, and
(b) How long does it take the object of tube A to reach the other
end? Assume there is no friction inside the tubes, and the density of the earth
is constant.
And the answer is:
(a)
The
time to traverse any such a tube is always the same. It takes approximately 42
minutes for an object to move from one end of the tube to the other,
irrespective of the location and length of the tube. This will be proved in
part (b).
(b)
Let
M
be the mass of the earth, m be the mass of the object, r
its distance from the center of the earth (at any time), and R
the radius of the earth. Also x is the distance of the object from
the center of the tube, at any time. The angle a is the angle between
the perpendicular line to the tube and r. All these variables are
represented in the following figure
The
force of gravitation pulling the object to the center of the earth is due to
the portion of mass of the earth inside the sphere of radius r.
The volume of this sphere is (remember that we assumed that earth density is
constant, so the mass is proportional to the volume)
The
force on the object is given by
To
calculate the speed of the object along the tube, we need the component of this
force along the tube. Assume that the object is moving from A to B, so
This
force is equal to the Newtonian expression for force, or
, or
Or,
where .
This
is the wellknown simple harmonic equation (who hasn't solve it? Solution: http://en.wikipedia.org/wiki/Simple_harmonic_motion) . The general
solution is given by
and
ω is the frequency of oscillation given by
Then,
the period of oscillation is
This
is a constant. The mass and radius of
the earth and the gravitational constants are known constants given by
Substituting
these values in the above equation we get
This
is the time for the object to produce a full oscillation; the time it takes to
the object to move oneway is

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