What is Simple harmonic oscillator: Definition and 114 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.
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...
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...
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...
We show by working backwards
$$\hbar w \Big(a^{\dagger}a+\frac{1}{2}\Big)=\hbar w \Big(\frac{mw}{2\hbar}(\hat{x}+\frac{i}{mw}\hat{p})(\hat{x}-\frac{i}{mw}\hat{p})+\frac{1}{2}\Big)$$...
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.
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...
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.
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...
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}...
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...
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 =...
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...
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 $$...
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...
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...
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...
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...
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'...
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...
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...
Homework Statement
An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion.
Homework Equations
##F=m\ddot{x}##
The Attempt at a Solution
We have...
Homework Statement
Show that the virial theorem holds for all harmonic-oscillator states. The identity given in problem 5-10 is helpful.
Homework Equations
Identity given: ∫ξ2H2n(ξ)e-ξ2dξ = 2nn!(n+1/2)√pi
P.S the ξ in the exponent should be raised to the 2nd power. So it should look like ξ2...
Homework Statement
Show that application of the lowering Operator A- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22).
Homework Equations
Equation (5.6.22): A-Ψn = -iΨn-1√n
The Attempt at a Solution
I began by saying what the answer should end...
Homework Statement
Two masses ##m_1## and ##m_2## are joined by a spring of spring constant ##k##. Show that the frequency of vibration of these masses along the line connecting them is:
$$\omega =\sqrt{\frac{k(m1+m2)}{m1m2}}$$
Homework Equations
##x(t)=Acos(\omega t)##
##\omega...
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...
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...
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...
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##...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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" =...
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...
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)...