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.
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...
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.
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 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...
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 $$...
I started off by finding when Fg=Fx:
(72)(x)=(31)(9.8)
x=4.2193m
After this I'm stuck and have a few things I'm confused about:
When the penguin's jumping, is there elastic energy? (because the rope's getting compressed? Or maybe not). Also, I know you can use energy conservation, but...
So I tried solving the differential equation for a spring - mass system using Euler's Algorithm in Python. The equation being
d2x/dt2= -4π2x
(The equation was obtained by Dimensional Analysis)
here x and t are both dimensionless equivalents of position...
Homework Statement
A vertical block-spring system on Earth has a period of 6.0 s. What is the period of this same system on the moon where the acceleration due to gravity is roughly 1/6 that of earth?
Homework Equations
w = √(k/m)
w = (2Pi)/T
T = 2Pi*√(m/k)[/B]
The Attempt at a Solution
So...
Homework Statement
My question here isn't a specific question that has been given for homework, but a more general one. For an assignment I have to 'derive an expression for the resonant frequency, ω0' for two different systems, the first for 'a mass M connected to rigid walls via two springs'...
Homework Statement
The amplitude of any oscillator can be doubled by:
A. doubling only the initial displacement
B. doubling only the initial speed
C. doubling the initial displacement and halving the initial speed
D. doubling the initial speed and halving the initial displacement
E. doubling...
A mass attached to a spring is oscillating in Simple Harmonic Motion. If an other spring of same sprinc constant is attached parrallel to the other spring, what is the period of this new system (as a function of the initial period).
Here's what I did and have no idea if this is right:
For the...
Homework Statement
An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion.
Homework Equations
##F=m\ddot{x}##
The Attempt at a Solution
We have...
Homework Statement
Two masses ##m_1## and ##m_2## are joined by a spring of spring constant ##k##. Show that the frequency of vibration of these masses along the line connecting them is:
$$\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##...
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)...
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...