# Simple harmonic oscillator Definition and 26 Discussions

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. ### 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...
2. ### 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...
3. ### 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?
4. ### Hamiltonian of a displaced QHO

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...
5. ### 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.
6. ### 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...
7. ### 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...

17. ### 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...
18. ### The simple pendulum

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...
19. ### 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...
20. ### 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##...
21. ### Is it SHM or not?

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...
22. ### 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...
23. ### 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...
24. ### 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)...
25. ### 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...
26. ### 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...