CP An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. ( Note: The gravitational force on the object as a function of the object’s distance r from the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration a x and the displacement from equilibrium x are related by Eq. (14.8), and the period is then T = 2 π / ω .)
CP An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. ( Note: The gravitational force on the object as a function of the object’s distance r from the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration a x and the displacement from equilibrium x are related by Eq. (14.8), and the period is then T = 2 π / ω .)
CP An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. (Note: The gravitational force on the object as a function of the object’s distance r from the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration ax and the displacement from equilibrium x are related by Eq. (14.8), and the period is then T = 2π/ω.)
Pr5. A pointlike body of mass m made of
lead is fixed inside a homogeneous solid sphere
of radius R and mass m at distance R/2 from
the center of the sphere. This body is placed on
a horizontal rough surface. Find the period of
small oscillations of the sphere around its equi-
librium position. (The sphere rolls without slip-
ping on the surface. The moment of inertia of a
homogeneous sphere of mass m and radius R is
2mR* /5.)
R/2
(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.)
(b)Write an equation that represents the curve in the form
The centroidal radius of gyration Ky of an airplane is determined by suspending the airplane by two 12-ft-long cables as shown. The airplane is rotated through a small angle about the vertical through G and then released. Knowing that the observed period of oscillation is 3.3 s, determine the centroidal radius of gyration Ky.
Chapter 14 Solutions
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