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Consider the following equation of motion for a damped driven harmonic oscillator: * + 1 +...
3. A damped harmonic oscillator is driven by an external force of the form mfo sin ot. The equation of motion is therefore x + 2ßx + ω x-fo sin dot. carefully explaining all steps, show that the steady-state solution is given by x(t) A() sin at 8) Find A (a) and δ(w).
2. Solve for the motion of a driven-damped harmonic oscillator whose forcing function F(t) is given by F(t) = 0, < 0 HU t 12 0 <t<T PO) – 4(), 0<t<7 F(t) = A, t>t. m т.
7. (a) Explain what is meant by damped harmonic motion, and write down a differential equation describing this phenomenon b) Give an example of a damped harmonic oscillator in practice. Sketch the oscilla- tions it undergoes, and calculate their frequency and damping rate for a natural (undamped) frequency wo 10 Hz and damping coefficient γ-: 2.0 s-1
7. (a) Explain what is meant by damped harmonic motion, and write down a differential equation describing this phenomenon b) Give an example...
. Consider a weakly damped oscillator that is being driven by the periodic rectangular pulses shown in the following figure. Let the natural period of the oscillator beTo-3 and the damping constant be β = 0.1. Let the pulse last for a time Δτ= 1.5 and have a height fmax -5. Calculate the first four Fourier coefficient An for the long-term motion x(t) of the oscillator, assuming that the relation between the drive period τ and the natural period τοΒ...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
2. A damped harmonic oscillator with m 1.00 kg, k 2500 N/m, and b 42.4 kg/s is subject to a driving force given by Fo cos wt. (a) what value of ω results in the maximum stead-state amplitude (ie, resonance)? (b) What is the quality factor Q of this oscillator?
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest at X +0.750 m. The oscillator is set into motion once again from x=+0.750 m, except the oscillator now experiences a retarding force that is linear with respect to velocity. As a result, the oscillator does not return to its original starting position, but instead reaches = +0.700 m after one period. a. During the first full oscillation of motion, determine the fraction of...
A Wave Packet in Simple Harmonic Motion: Coherent State of Simple Harmonic Oscillator 2 Background: Without the general tools for solving the Time Dependent Schrödinger Equation DSwhich we will lear in ciect ssoltions io the TDSEi are diflieli but not impossible to find. In this problem, you will consider one such solution, the "Coherent States" of a Simple Harmonic Oscillator (SHO) of frequency w. We will use the solution to this problem to illustrate the general principles of the Correspondence...
For the harmonic oscillator with mel, C=8, K:16 ; Xo = 5, Vo=4, do the following a. Find the position function x(t) and determine whether the motion is overdamped, critically damped or underdamped. b. Find the undamped (when c-0) position function ult) = Co Cos(wot-do)
Problem 15. (20 pts) Consider a damped driven oscillator with the following parameters s-100 N/m b=0.5kg/s m= 1 kg Fo=2N A) Find the resonant frequency, w. B) Find the damping rate y C) What is the quality factor Q for this oscillator? D) Is this oscillator lightly damped, critically damped, or heavily damped? E) Find the steady state amplitude when the oscillator is driven on resonance (Ω=w). F) Find the steady state amplitude when Ω_w+γ/2. G) Find the average power...