Write down the solution to the oscillator equation for the case ω0>γ if the oscillator starts...
Write down the solution to the oscillator equation for the case ω0>γ if the oscillator starts from x=0 with velocity v. Show that, as ω0 is reduced to the critical value γ, the solution tends to the corresponding solution for the critically damped oscillator.
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Write down the solution to the oscillator equation for the case
ω0>γ if the oscillator starts...
A simple damped mechanical harmonic oscillator with damping constant γ is driven by a force ?0?????....
A simple damped mechanical harmonic oscillator with damping constant γ is driven by a force ?0?????. Show that the FWHM of the amplitude A(ω) vs. angular frequency ω curve is ?√3. You can assume that Q>>1 and ω is very close to ω0. Formulae in the book can be used. But you will have to reference the page and equation number.
Each of the following equation represents an unforced damped oscillator. Write the Laplace transform of the...
Each of the following equation represents an unforced damped oscillator. Write the Laplace transform of the characteristic equation. And define if the system is over-damped, under-damped or critically damped. 1) 23 + 4 + 2x = 0 2) 3 +43 +3.2 = 0 3) 43 + 7 + 5x = 0 BIU A - A - IX E 33 x X, DE EV G* T 1 12pt Paragraph
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 os...
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...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a box with mass m attached to a Hook's spring of coefficient k with linear air friction of coefficient n) is described by mx"(t) + n2'(t) + ku(t) = 0 where m, n, k > 0. (a) Write down the solutions for three cases and their long term limits 1. Overdamped: when friction is strong 1 > 4mk 2. Underdamped: when friction is weak 72...
1) Answer the following questions for harmonic oscillator with the given parameters and initial c...
1) Answer the following questions for harmonic oscillator with the given parameters and initial conditions Find the specific solution without converting to a linear system Convert to a linear system Find the eigenvalues and eigenvectors of the corresponding linear system Classify the oscillator (underdamped, overdamped, critically damped, undamped) (use technology to) Sketch the direction field and phase portrait Sketch the x(t)- and v(t)-graphs of the solution a. b. c. d. e. f. A) mass m-2, spring constant k 1, damping...
is a continuation of problem 5.49.) 5) The Green's function for a linear oscillator that starts...
is a continuation of problem 5.49.) 5) The Green's function for a linear oscillator that starts from rest is e-pe-e') sin((t- t)), for t2t G(t, t') =ma, 0, for t<t The solution for a forced oscillator with a forcing function F(t) is then x(t) F(t')G(t,&')d¢ 00 a) (3 pts) Calculate x(t) for an oscillator for the case where it is undamped, has natural frequency oo, and is driven by the following force function: it is zero before t 0, is...
Using Octave to solve (preferably with solving the differential equations and go through the process) 1. A harmonic osc...
Using Octave to solve (preferably with solving the differential
equations and go through the process)
1. A harmonic oscillator obeys the equation dx dt dt which can be written as a set of coupled first order differential equations dx dt dt One procedure in Octave for coding these equations involves a global statement and the line solutionRC Isode(@dampedOscillator, [1, 0], timesR); Employ the help system to determine the properties of the Isode() function (or an equivalent solver such as ode23()...
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiabl...
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiable) plane curve. For curves such that lh'(t) 0, the (signed) curvature is defined to be the quantity K(t) (a) Suppose the curve γ(t) is the graph of a function, ie x(t)-t and y(t) f(t) for some function f. Write the formula for the curve in this case. Suppose you were at a critical point of the graph of f. What does the curvature...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
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...
3. Power in a DHO. The power developed in a mechanical system is force x velocity. Show that the ...
3. Power in a DHO. The power developed in a mechanical system is force x velocity. Show that the average power dissipated by the damping in a DHO is: (b) Evaluate this expression at o r, the resonant frequency from Problem 2. Show that the average driving power, <Pin is: (Hint: The average of cos2@t-ф) is %.) Useful Formulae Natural Frequency co-sqrt(km); ω0-2to; simple harmonic oscillator: d2x/dt2 + 002x-0; Forced Damped Harmonic Oscillator (DHO): m"(d2x/dt2) + b*(ds/dt) + kx =...
Each of the following equation represents an unforced damped oscillator. Write the Laplace transform of the characteristic equation. And define if the system is over-damped, under-damped or critically damped. 1) 23 + 4 + 2x = 0 2) 3 +43 +3.2 = 0 3) 43 + 7 + 5x = 0 BIU A - A - IX E 33 x X, DE EV G* T 1 12pt Paragraph
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...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a box with mass m attached to a Hook's spring of coefficient k with linear air friction of coefficient n) is described by mx"(t) + n2'(t) + ku(t) = 0 where m, n, k > 0. (a) Write down the solutions for three cases and their long term limits 1. Overdamped: when friction is strong 1 > 4mk 2. Underdamped: when friction is weak 72...
1) Answer the following questions for harmonic oscillator with the given parameters and initial conditions Find the specific solution without converting to a linear system Convert to a linear system Find the eigenvalues and eigenvectors of the corresponding linear system Classify the oscillator (underdamped, overdamped, critically damped, undamped) (use technology to) Sketch the direction field and phase portrait Sketch the x(t)- and v(t)-graphs of the solution a. b. c. d. e. f. A) mass m-2, spring constant k 1, damping...
is a continuation of problem 5.49.) 5) The Green's function for a linear oscillator that starts from rest is e-pe-e') sin((t- t)), for t2t G(t, t') =ma, 0, for t<t The solution for a forced oscillator with a forcing function F(t) is then x(t) F(t')G(t,&')d¢ 00 a) (3 pts) Calculate x(t) for an oscillator for the case where it is undamped, has natural frequency oo, and is driven by the following force function: it is zero before t 0, is...
Using Octave to solve (preferably with solving the differential
equations and go through the process)
1. A harmonic oscillator obeys the equation dx dt dt which can be written as a set of coupled first order differential equations dx dt dt One procedure in Octave for coding these equations involves a global statement and the line solutionRC Isode(@dampedOscillator, [1, 0], timesR); Employ the help system to determine the properties of the Isode() function (or an equivalent solver such as ode23()...
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiable) plane curve. For curves such that lh'(t) 0, the (signed) curvature is defined to be the quantity K(t) (a) Suppose the curve γ(t) is the graph of a function, ie x(t)-t and y(t) f(t) for some function f. Write the formula for the curve in this case. Suppose you were at a critical point of the graph of f. What does the curvature...
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...
3. Power in a DHO. The power developed in a mechanical system is force x velocity. Show that the average power dissipated by the damping in a DHO is: (b) Evaluate this expression at o r, the resonant frequency from Problem 2. Show that the average driving power, <Pin is: (Hint: The average of cos2@t-ф) is %.) Useful Formulae Natural Frequency co-sqrt(km); ω0-2to; simple harmonic oscillator: d2x/dt2 + 002x-0; Forced Damped Harmonic Oscillator (DHO): m"(d2x/dt2) + b*(ds/dt) + kx =...
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