Simple harmonic oscillator Definition and 114 Threads

In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle approximation). Simple harmonic motion can also be used to model molecular vibration as well.
Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of Fourier analysis.

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  1. J

    Modification to the simple harmonic oscillator

    I was assuming there could be something via perturbation theory? I am unsure.
  2. S

    I How to interpret complex solutions to simple harmonic oscillator?

    Consider the equation of motion for a simple harmonic oscillator: ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
  3. Misha87

    B Harmonic oscillator and simple pendulum time period

    Hi, I have been thinking about pendulums a bit and discovered that a HO(harmonic Oscillator) will take the same time to complete one period T no matter which amplitude A/length l it has, if stiffness k and mass m are the same. But moving on to a simple pendulum suddenly the time period for one...
  4. Huzaifa

    B Why is a simple pendulum not a perfect simple harmonic oscillator?

    Khan Academy claims that a simple pendulum not a perfect simple harmonic oscillator. Why is it so?
  5. tanaygupta2000

    How Do You Operate the Hamiltonian on a Coherent State?

    I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is...
  6. D

    Simple harmonic oscillator Hamiltonian

    We show by working backwards $$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$...
  7. T

    Does change in 'g' affect frequency of mass spring system?

    I attempted using f = 1/(2pi x sqrt l/g) For Earth I found the value of length to be 0.0276m. Then I substituted the value in the equation, putting (1/3)g instead of g, to find the value of f in Mars. My answer is C. I am confused. Please help me.
  8. A

    Hello Reality Anyone familiar with the Davisson-Germer Experiment?

    Greetings, I'm happy to find such an enthusiastic community with an encyclopedic knowledge and mathematical rigor. I'm a Biomedical Engineering Researcher that's had to breach into the world of condensed matter physics to better understand the physical principles of the piezoelectric crystal...
  9. M

    A critically damped simple harmonic oscillator - Find Friction

    c = Critically Damped factor c = 2√(km) c = 2 × √(150 × .58) = 18.65 Friction force = -cv Velocity v = disp/time = .05/3.5 Friction force = - 18.65 * .05/3.5 = -.27 N I am not sure if above is correct. Please check and let me know how to do it.
  10. Vivek98phyboy

    I Why is this SHM the way it is?

    I know four different forms in which an SHM can be represented after solving the differential and taking the superposition acos(wt+Ø) asin(wt+Ø) acos(wt-Ø) asin(wt-Ø) where a- amplitude In the above image they took B as negative in order to arrive at acos(wt+e). If i already knew i wanted...
  11. T

    Simple Harmonic Oscillator Squeezing

    I'm working through https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_06.pdf, and I'm stumped how they got from Equation 5.26 (##\vert 0_{\gamma} \rangle \equiv \frac{1}{\sqrt{cosh\gamma}} exp(-\frac{1}{2}tanh\gamma \hat{a^\dagger}\hat{a^\dagger}...
  12. I

    Time Derivatives of Expectation Value of X^2 in a Harmonic Oscillator

    I can show that ##\frac{d}{dt} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{1}{m} \langle \psi (t) \vert PX+XP \vert \psi (t) \rangle##. Taking another derivative with respect to time of this, I get ##\frac{d^2}{dt^2} \langle \psi (t) \vert X^2 \vert \psi (t) \rangle = \frac{i}{m...
  13. M

    MATLAB No damping but the solution to simple harmonic oscillator damps?

    I posted yesterday but figured it out; however, a different issue I just detected with the same code arose: namely, why does the solution damp here for an undamped simple harmonic oscillator? I know the exact solution is ##\cos (5\sqrt 2 t)##. global delta alpha beta gamma OMEG delta =...
  14. Glenn Rowe

    A Feynman propagator for a simple harmonic oscillator

    I'm reading through Lancaster & Blundell's Quantum Field Theory for the Gifted Amateur and have got to Chapter 17 on calculating propagataors. In their equation 17.23 they derive the expression for the free Feynman propagator for a scalar field to be...
  15. M

    Simple harmonic motion of a bar pivoted at one end

    Hi, I am unsure how to proceed with this problem. I believe that I can correctly calculate the frequency of the oscillations for a bar that is not suspended from a spring but I do not know how to take the effect of the spring into account. The answer given by my professor is $$...
  16. bigbosswilly

    High school Physics - Simple Harmonic Motion

    I started off by finding when Fg=Fx: (72)(x)=(31)(9.8) x=4.2193m After this I'm stuck and have a few things I'm confused about: When the penguin's jumping, is there elastic energy? (because the rope's getting compressed? Or maybe not). Also, I know you can use energy conservation, but...
  17. Dr. Courtney

    Insights An Accurate Simple Harmonic Oscillator Laboratory - Comments

    Greg Bernhardt submitted a new blog post An Accurate Simple Harmonic Oscillator Laboratory Continue reading the Original Blog Post.
  18. TachyonLord

    The order of calculating velocity and position alters the solution?

    So I tried solving the differential equation for a spring - mass system using Euler's Algorithm in Python. The equation being d2x/dt2= -4π2x (The equation was obtained by Dimensional Analysis) here x and t are both dimensionless equivalents of position...
  19. J

    Confused about a simple harmonic motion problem....

    Homework Statement A vertical block-spring system on Earth has a period of 6.0 s. What is the period of this same system on the moon where the acceleration due to gravity is roughly 1/6 that of earth? Homework Equations w = √(k/m) w = (2Pi)/T T = 2Pi*√(m/k)[/B] The Attempt at a Solution So...
  20. B

    Simple Harmonic Oscillator with Boundary Conditions

    How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...
  21. B

    Derivation of resonant frequency for SHM systems

    Homework Statement My question here isn't a specific question that has been given for homework, but a more general one. For an assignment I have to 'derive an expression for the resonant frequency, ω0' for two different systems, the first for 'a mass M connected to rigid walls via two springs'...
  22. E

    How can we double the amplitude of an oscillator?

    Homework Statement The amplitude of any oscillator can be doubled by: A. doubling only the initial displacement B. doubling only the initial speed C. doubling the initial displacement and halving the initial speed D. doubling the initial speed and halving the initial displacement E. doubling...
  23. S

    Period of a mass spring system with 2 spring of same K(vert)

    A mass attached to a spring is oscillating in Simple Harmonic Motion. If an other spring of same sprinc constant is attached parrallel to the other spring, what is the period of this new system (as a function of the initial period). Here's what I did and have no idea if this is right: For the...
  24. V

    Exponentially driven harmonic oscillator

    Homework Statement An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion. Homework Equations ##F=m\ddot{x}## The Attempt at a Solution We have...
  25. N

    Virial Theorem and Simple Harmonic Oscillator

    Homework Statement Show that the virial theorem holds for all harmonic-oscillator states. The identity given in problem 5-10 is helpful. Homework Equations Identity given: ∫ξ2H2n(ξ)e-ξ2dξ = 2nn!(n+1/2)√pi P.S the ξ in the exponent should be raised to the 2nd power. So it should look like ξ2...
  26. N

    Lowering Operator Simple Harmonic Oscillator n=3

    Homework Statement Show that application of the lowering Operator A- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22). Homework Equations Equation (5.6.22): A-Ψn = -iΨn-1√n The Attempt at a Solution I began by saying what the answer should end...
  27. flamespirit919

    Frequency of Vibration of Two Masses Joined by a Spring

    Homework Statement Two masses ##m_1## and ##m_2## are joined by a spring of spring constant ##k##. Show that the frequency of vibration of these masses along the line connecting them is: $$\omega =\sqrt{\frac{k(m1+m2)}{m1m2}}$$ Homework Equations ##x(t)=Acos(\omega t)## ##\omega...
  28. R

    Simple Harmonic Oscillator behaviour when a potential term is added

    Homework Statement A simple harmonic oscillator has a potential energy V=1/2 kx^2. An additional potential term V = ax is added then, a) It is SHM with decreased frequency around a shifted equilibrium b) Motion is no longer SHM c)It is SHM with decreased frequency around a shifted equilibrium...
  29. SebastianRM

    Simple Pendulum: Understand the Relationship Between Theta & L

    1. Homework Statement Hey guys, I am reading my Physics book, in that specific section it says "the restoring force must be directly proportional to x or (because x=(theta)*L) to theta" Homework Equations The Attempt at a Solution I have tried to look for that x=(theta)*L relationship...
  30. Wes Ellgass

    Simple harmonic motion calculations from doubling the mass.

    Homework Statement What will the new amplitude be if A=.117m and the mass is 0.1kg. The spring constant is 3.587N/m and the mass is then doubled. What is the new velocity max? What is the acceleration max? Homework Equations Fnet= -kx, vmax=A(ω), ω= √k/m The Attempt at a Solution...
  31. koustav

    I Exploring the 1/2 Factor in Simple Harmonic Oscillator Solutions

    In the series solution of simple harmonic oscillator,why do we have a factor of 1/2 in the trial solution?
  32. J

    I Solving for SHM Diatomic Energy Levels

    So I'm trying to figure out how we got the allowed vibrational energy levels for a diatomic molecule by approximating it with simple harmonic motion. I do know how to use the uncertainty principle to get the zero-point energy: We know that the potential function is ##V(x) = \frac{1}{2}mx^2##...
  33. Ethan Godden

    Frequency of a simple harmonic oscillator

    Homework Statement The problem is attached Homework Equations f=2π/ω=2π√(m/k) The Attempt at a Solution My idea is that the mass doubles resulting in a √2 increase in the equation above. However, apparently the answer is (c). I have a strong feeling the book answer is wrong, but I wanted to...
  34. Ethan Godden

    Time period for a Simple Harmonic Oscillator to go from 0-1m

    Homework Statement A particle with a mass(m) of 0.500kg is attached to a horizontal spring with a force constant(k) of 50.0N/m. At the moment t=0, the particle has its maximum speed of 20m/s and its moving to the left. Find the minimum time interval required for the particle to move from...
  35. S

    Is a Vertically Hung Spring Mass System SHM?

    In an SHM, the only force that should be acting, that is the net force should be the restoring force F, by definition... F = -kx For example there is a massless spring of spring constant k attached to the ceiling and there is a body of mass m hung at it and avoiding all kinds of friction...
  36. tridianprime

    Simple Harmonic Motion Average Velocity

    Homework Statement At time t = 0, a point starts oscillating on the x - axis according to the law x = a sin(ωt). Find the average velocity vector projection (I assume it means magnitude based on previous questions in the book). Homework Equations The Attempt at a Solution I knew that the...
  37. A

    Total Energy of a movable pivot-pendulum system, and ω

    Homework Statement This is not really a homework questions, just part of my notes confusing me a bit. This is the derivation of total energy for a pendulum of mass m2 with movable pivot of mass m1. I don't understand how frequency can be read off. What am I missing? Homework Equations See...
  38. Alettix

    Velocity when falling into planet

    Homework Statement Hi! I would need a little help with the following problem: We have found a new planet with density ρ and radius R, and drill a hole to its center. Then accidentally, one person falls into the hole. What is his velocity when reaching the bottom (the center of the planet)...
  39. N

    Driving force in simple harmonic oscillator with exponential

    Homework Statement A particle in SHM is subject to a driving force F(t)= ma*e^(-jt). Initial position and speed equal 0. Find x(t). Homework Equations F = -kxdx = mvdv F(t) = F(0)*e^(iωt) x(t) = Acos (ωt +φ) The Attempt at a Solution I have no idea how to deal with the exponential term. I...
  40. S

    Simple Harmonic Oscillator Zero Probability Points

    Hi, What is the physical meaning of zero probability of finding a particle in the square of the Quantum SHO wave function? the particle is supposed to oscillate about the equilibrium position, how would it go from an end point to the other end point without passing by certain points? Could the...
  41. T

    Simple Harmonic Motion with Damping and Driving

    Hello, I was asked by my professor today to graph the motion, as well as the energies, of a spring that undergoes driven and/or damped oscillation; however, I was unable to because I do not have a very good idea of how they work. Can someone explain to me, qualitatively, what it means to have a...
  42. Futurestar33

    Given a harmonic oscillator with mass m, and spring constant

    Homework Statement Given a harmonic oscillator with mass m, and spring constant k, is subject to damping force F= cdx/dt and driven by an external force of the form F[ext]= FoSin(wt). A) Find the steady state solution. B) Find the amplitude and the phase. Homework Equations F=-kx the steady...
  43. Greg Bernhardt

    What is a simple harmonic oscillator

    Definition/Summary An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time. Equations x(t)=A\sin(\omega t)+B\cos(\omega t) \omega^2 =\frac{k}{m} Extended explanation According to...
  44. E

    Simple Harmonic Oscillator on a smooth surface

    I feel I understand what happens, and how to solve the equation of motion x(t) for a mass attached to a spring and released from rest horizontally on a smooth surface. We typically end up with x(t) = x_0 cos(ωt) as the solution, with x_0 as the amplitude of the oscillation. But I've...
  45. Maxo

    Changing the mass of a simple harmonic oscillator

    Homework Statement Homework Equations The Attempt at a Solution I find this task very hard to understand. First of all, when adding more mass, wouldn't that change the acceleration, according to F=ma? And in that case the velocity should also change when adding more mass, shouldn't it? That...
  46. M

    A simple harmonic oscillator has total energy E= ½ K A^2

    A simple harmonic oscillator has total energy E= ½ K A^2 Where A is the amplitude of oscillation.  E= KE+PE a) Determine the kinetic and potential energies when the displacement is one half the amplitude. b) For what value of the displacement does the kinetic energy equal the potential...
  47. ShayanJ

    MATLAB Troubleshooting Simple Harmonic Oscillator in MATLAB

    I'm trying to plot the evolution of a simple harmonic oscillator using MATLAB but I'm getting non-sense result and I have no idea what's wrong! Here's my code: clear clc x(1)=0; v(1)=10; h=.001; k=100; m=.1; t=[0:h:10]; n=length(t); for i=2:n F(i-1)=-k*x(i-1)...
  48. M

    Hamiltonian For The Simple Harmonic Oscillator

    I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" =...
  49. A

    Number of States in a 1D Simple Harmonic Oscillator

    Homework Statement A system is made of N 1D simple harmonic oscillators. Show that the number of states with total energy E is given by \Omega(E) = \frac{(M+N-1)!}{(M!)(N-1)!} Homework Equations Each particle has energy ε = \overline{h}\omega(n + \frac{1}{2}), n = 0, 1 Total energy is...
  50. L

    Simple Harmonic Oscillator and Damping

    Homework Statement After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of it's initial value. Find the ratio of the frequency of this oscillator to that of it's natural frequency (undamped value) Homework Equations x'' +(√k/m) = 0 x'' = d/dt(dx/dt)...
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