James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1182 chapter 17 Second-Order Differential Equations
20. A spring with a mass of 2 kg has damping constant 16, and a
force of 12.8 N keeps the spring stretched 0.2 m beyond its
natural length. Find the position of the mass at time t if it
starts at the equilibrium position with a velocity of 2.4 mys.
21. Assume that the earth is a solid sphere of uniform density with
mass M and radius R − 3960 mi. For a particle of mass m
within the earth at a distance r from the earth’s center, the
gravi tational force attracting the particle to the center is
F r − 2GMrm
r 2
where G is the gravitational constant and M r is the mass of the
earth within the sphere of radius r.
(a) Show that F r − 2GMm
R 3 r.
(b) Suppose a hole is drilled through the earth along a diameter.
Show that if a particle of mass m is dropped from rest
at the surface, into the hole, then the distance y − ystd of
the particle from the center of the earth at time t is given by
y0std − 2k 2 ystd
where k 2 − GMyR 3 − tyR.
(c) Conclude from part (b) that the particle undergoes simple
harmonic motion. Find the period T.
(d) With what speed does the particle pass through the center
of the earth?
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