For the cylinder of Prob. 8 / 10, determine the vertical...

For the cylinder of Prob. $8 / 10$, determine the vertical displacement $x,$ measured positive down in millimeters from the equilibrium position, in terms of the time $t$ in seconds measured from the instant of
release from the position of $25 \mathrm{mm}$ added deflection.

Key Concepts

Simple Harmonic Motion

This concept describes the type of motion where the restoring force is directly proportional to the displacement from an equilibrium position and is directed towards that equilibrium. The motion is periodic and can be modeled by a second–order linear differential equation whose solutions are sine and cosine functions.

Equation of Motion for Oscillatory Systems

The displacement of a system undergoing simple harmonic motion is typically expressed in the form x(t) = A cos(?t + ?) or x(t) = A sin(?t + ?), where A is the amplitude, ? is the angular frequency, and ? is the phase shift. This formulation encapsulates the time–dependent behavior of the system.

Angular Frequency

Angular frequency, denoted by ?, quantifies how rapidly the system oscillates and is determined by the physical parameters of the system, such as mass and stiffness (or spring constant). It appears in the differential equation and hence in the trigonometric solution of the system's motion.

Initial Conditions

Initial conditions, such as the initial displacement and velocity, are used to uniquely determine the constants in the general solution of the differential equation. These conditions allow one to calculate specific values for the amplitude and phase shift, ensuring the solution accurately reflects the initial state of the system.

Phase Shift

The phase shift (or phase angle) in the solution of a harmonic oscillator accounts for any initial offset in the cycle of motion. It is determined by applying the initial conditions to the general trigonometric solution, thereby aligning the mathematical model with the actual motion that starts from a specific displaced position.

Transcript

00:01 Hello, we're going to do a continuation of the previous problem that i answered on number 10.

00:06 So if you need to see how we went through that, please refer to that answer.

00:11 This problem asks for the equation of motion of the position of this mass and spring system shown in the red box on the left.

00:20 So we went through before and found the spring constant and we found the resonance frequency of this system to be 2 .23 hertz.

00:29 Now we want to know what is the equation as a function of time for this system.

00:36 And additionally, we know that the system starts with an additional displacement so that it's at 75 millimeters, and that we know it starts at a velocity of zero meters per second.

00:53 That was given in the problem.

00:55 The question is, what is the position equation as a function of time? if you refer to the text, we can start from the base definition, where x of t is going to be defined as x -not cosine omega -t plus x.

01:15 Dot not over omega -sign omega -t.

01:19 This is the definition of a harmonic oscillator like this problem and a spring in a mass that has some position, initial position, and initial velocity.

01:30 Now we need to get omega.

01:33 We do not have that, but we do know the resonance frequency, so we can easily get omega.

01:38 So recall that omega, the angular frequency, is equal to 2 pi times that resonance frequency.

01:51 So that's just a simple multiplication, 2 pi times 2 .23 hertz...

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