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.)
When setting the length of the pendulum, it is important that:
The length of string from the pivot to the surface of the ball is exactly the stated
length for the case.
The length of string from the pivot to the center of the ball is exactly the stated length
for the case.
The accurately measure length of the string plus the radius of the ball is used in the
calculations.
A solid cylinder of mass M, radius R, and length L, rolls around a light, thin axle at its center. The axle is attached to a spring of stiffness K on either side. The springs are attached to same wall. The cylinder is brought back a distance A, parallel to the wall, and allowed to oscillate. What is the maximum speed of the oscillations of the cylinder?
Answer: ((3*k*A^2)/(2*M))^1/2
MärkxsfH)
Problem 1:
A mass of 20 grams stretches a spring by 10/169 meters.
(a) Find the spring constant k, the angular frequency w, as well as the
period T and frequency f of free undamped motion for this spring-mass
system.
(b) Find the general solution x(t) of the DE for the free spring-mass
system.
(c) Suppose that an exterior force of
F(t) = 2.5 sin(12t) in Newtons
acts on the spring-mass system. Find the equation of motion (the solution
x(t)) of the system if the mass initially is at rest in its equilibrium position.
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