For the system of Prob. 8 / 2, determine the position x as a function of time if the mass is rel

For the system of Prob. 8 / 2, determine the position x as a...

Key Concepts

Phase Shift

The phase shift in a sinusoidal function accounts for the initial conditions of the system, such as the initial displacement and velocity. It adjusts the starting point of the oscillation in time, ensuring that the mathematical model accurately reflects the system’s state at time zero.

Amplitude

Amplitude represents the maximum displacement of the oscillating system from its equilibrium position. It determines the overall extent of the motion and is a crucial parameter in defining the energy and the peak values in the sinusoidal representation of the displacement.

Period

The period is the duration of one complete cycle of oscillation. It is inversely related to the angular frequency by the relation ? = 2?/? and provides a measure of the time scale over which the oscillatory motion repeats itself.

Angular Frequency

Angular frequency (often denoted as ?) is a key parameter in simple harmonic motion that represents how rapidly the system oscillates. It is related to the physical properties of the system, such as mass and stiffness, and appears in the sinusoidal function as the coefficient of time.

Simple Harmonic Motion

This is the type of motion experienced by systems like mass-spring oscillators, characterized by oscillations about an equilibrium position that can be modeled using sinusoidal functions. The system’s displacement is typically expressed as a cosine or sine function of time, indicating a regular, repeating motion under a restoring force proportional to the displacement.

Transcript

00:02 Hello, here again we are going to be continuing from a previous problem.

00:06 Please see my previous answers to get all the details, but we are still discussing this system of a spring attached to a mass and the motion of this particular system given some initial conditions.

00:19 Now we found previously that the angular velocity of this or the resonance frequency of this system is 18 radiance per second.

00:28 We will need that and then related to that the frequency is 2 .86 we will also need that the question now relates to how does the amplitude of this system and phase fee of the system change for a different set of initial conditions and those initial conditions are that the position at time zero is equal to now two positive two inches we typically in this text refer to that as x not and the velocity, the x dot, the derivative of x, sorry, is at times zero, is equal to negative 9 because it says it's moving at 9 inches per second to the left, and we have defined the left as the negative direction.

01:21 Now, if you look at around equation 8 .5 to 8 .7 in this text, you will see that we can write the position in the 5.

01:32 Following form, some amplitude times sine of omega t plus some phase fee.

01:40 This is equivalent to the previous form that we had used in the last problem, something like this from equation 8 .6.

01:48 They're identical.

01:49 They're just useful for different perspectives.

01:51 In this problem, we are going to use this particular form.

01:56 Now, one of the things that the text also does is it tells us that there's a general relationship for c, which is that it is equal to x -not, the initial condition at times zero squared, plus the velocity x -not dot over the angular frequency squared, square -rooted.

02:16 So if we plug in all of the initial parameters up here, this is our x -not dot, just for reference.

02:26 If we plug those in, we end up getting that this is equal to 2 .06 inches.

02:32 So that's part of the question was just what is c...

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