What is Harmonic oscillation: Definition and 60 Discussions
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
F
→
=
−
k
x
→
,
{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
This question is based on the calculations in these notes on 2nd order unit step response.
Some Initial Observations
The scenario modeled here is an undamped spring-mass system that is at rest until time ##0##, at which point a constant force starts to act on the mass.
The force is finite and...
##v(t)=-a\omega\sin{(\omega t-\phi)}##
##x(0)=a\cos{(-\phi)}=x_0##
##v(0)=-a\omega\sin{(-\phi)}=v_0##
##\implies \tan{(-\phi)}=-\frac{v_0}{\omega x_0}##
##\implies \phi=-\tan{\left (-\frac{v_0}{\omega x_0}\right )}##
The solution to this problem says that we can find...
[Mentor Note: Two duplicate threads merged...]
in container with dimensions L×D, rests water of height H and density ρ. we disturb the water along L dimension, and accept an oscillation is caused on the free surface of the water, which maintains its flatness, so that the central of mass of the...
Hi,
so of course Φ0 = 15° and after solving after solving Φ(t=5*T = 5/f) I found γ = 0.012
I need help with b).
If I do 2° = 15° * exp(-0.012t)*cos(2πf*t), I'm not able to find t so I did something else by assuming that the amplitude decreases at a constant rate:
After 5*T = 5*1/f = 18.52 s...
Consider the following thought experiment...
You are an engineer with a very peculiar assignment. With a mind to reduce the investment of excessive human labor and material waste, you have been asked to build an apparatus similar to an old-fashioned voicepipe. You are asked to ensure that the...
It is true that at resonance frequency the phase-shift between input and output is 90 degrees, so my mind would think that this is ok. But I am kind of unsure because of the whole dividing by zero part.
If this isn't allowed: is there any way to calculate/measure the damping coefficient with...
First of all, I found a function of the distance of the object form the equivalence point in both cases. I got something like d=2d' where d is the distance at the first case and d' at the second. I did that because I wanted to find the frequency, and so first I need to find the period of...
Homework Statement
Harmonically fluctuating object. It`s full energy (E) is 3*10-5 J. Maximum force (F) on object is 1.5 * 10-3N. Period is 2 seconds (T) and starting phase (ƒ) is 60°. Need to write equation for these fluctuations.
E = 3*10-5 J
F= 1.5 * 10-3N
T = 2 s
ƒ = 60°
Homework Equations...
Homework Statement
The question is similar to last week’s, except that we will consider how friction may damp the oscillation with time. A block with mass m shown in the drawing is acted on by a spring with spring constant k. The block is pulled distance [x[/0] from equilibrium position (x=0)...
Homework Statement
The ground state energy of 5 identical spin 1/2 particles which are subject to a one dimensional simple harkonic oscillator potential of frequency ω is
(A) (15/2) ħω
(B) (13/2) ħω
(C) (1/2) ħω
(D) 5ħω
Homework Equations
Energy of a simple harmonic oscillator potential is
En...
Hello,
in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...
Homework Statement
Is the time average of the tension in the string of the pendulum larger or smaller than
mg? By how much?
Homework Equations
$$F = -mgsin\theta $$
$$T = mgcos\theta $$
The Attempt at a Solution
I'm mostly confused by what it means by time average. However from my...
For the harmonic oscillator, I'm trying to study qualitative plots of the wave function from the one-dimensional time independent schrodinger equation:
\frac{d^2 \psi(x)}{dx^2} = [V(x) - E] \psi(x)
If you look at the attached image, you'll find a plot of the first energy eigenfunction for...
Homework Statement
[/B]
For differential equation of the form
## y''- y = 0 ##
BC is
## y(1) = B ##
which usually have general solution
## y(x) = C1 e^x + C2 e^{-x} ##
But this manual I am reading always want to go with general solution
## y = C1 \cosh(x) + C2 \sinh( x) ##
I assume...
In school we have numerous exercises that ask you to find the time when a body passes a certain point for the nth time in simple harmonic oscillation. But it is a bit mentally taxing to solve with the actual formula of x=Asin(ωt + φ), just because you have to sort out all the infinite solutions...
Homework Statement
Hello all,
I have a question regarding the damping constant for a model of a vertically oscillating mass on a spring. I have read through one or two similar questions on this site but I think I can manage to be a little more specific about what I'm asking.
I am in a physics...
Homework Statement
An object of mass m = 300g is attached to a spring with a constant k = 3.0Nm-1 and is at rest on a smooth horizontal floor in a fluid where the resistive force is assumed to be linearly proportional to the velocity v. the object is then displaced 10mm to the right of the...
Homework Statement
...when she pulls the ball down 2.5cm from equilibrium and releases it from rest, it oscillates at 5.5 Hz. What is displacement y as functions of t?
Homework Equations
Y= Acos(omega t+phi)
The Attempt at a Solution
I'm almost certain I should instead be using sin to...
Hello, I encountered a mass on a string problem in which the mass, moved from the equilibrium, gets a harmonic motion. The catch, however, is that the mass of the string is not neglected. On the lecture, the prof. wanted to calculate, for some reason, the complete kinetic energy of the system...
Homework Statement
The 900-mg balance wheel of a certain clock is made up of a thin metal ring of radius 12 mm connected by spokes of negligible mass to a fine suspension fiber as in (Figure 1) . The back-and-forth twisting of the fiber causes the wheel to move in simple harmonic motion with...
Homework Statement
A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of oscillator is
A. ½kT
B. kT
C. ³⁄₂kT
D. 3kT
E. 6kT
Homework Equations
Equipartition theorem
The Attempt at a Solution
So I know the...
Homework Statement
A uniform rod of mass m and length L is freely pivoted at one end. What is the period of its oscillations? Icm for a uniform rod rotating about its centre of mass is 1/12mL2
(a) √3g/2L
(b) 2π √3L/2g
(c) 2π √2L/3g
(d) 2π √L/g
(e) none of the above
Homework Equations
ω2 =...
Homework Statement
Hi everybody!
Two masses m1 and m2 are connected with a spring one after the other to a wall (see attached picture). The spring constants are k1 and k2. To consider here are only longitudinal oscillations and no external forces.
a) Express the Newtonian equations of motion...
(a) A wave traveling on a Slinky® that is stretched to 4.5 m takes 2.6 s to travel the length of the Slinky and back again. What is the speed of the wave?
For this one, I did v = d/t
= 4.5 m / 2.6 s
= 1.73 m/s
Then I did v = (1.73)(2) = 3.46 m/s
This is correct
(b) Using the same Slinky...
Homework Statement
Compare the simple harmonic motion of two identical masses oscillating up and down on springs with different spring constants.
Homework Equations
F = -kxThe Attempt at a Solution
Okay, so I understand that the higher the spring constant, the harder it is to compress the...
Hello!
An assignment for my computational modeling course is to demonstrate the use of the Standard Euler method for modeling a simple harmonic oscillator; in this case, a mass attached to the end of a spring.
I have the two coupled first-order differential equations satisfying hookes law...
Hello,
When I have the differential equation
\frac{dY(x)}{dx} = -k^2 Y(x)
The solution is of course harmonic oscillation, however, looking at various places I see the solution given as:
Y(x) = A cos(kx) + B sin(kx)
instead of
Y(x) = A cos(kx + \phi_1) + B sin(kx + \phi_2)
Isnt...
Homework Statement
A block with a mass M is located on a frictionless, horizontal surface and is attached to a horizontal spring with spring stiffness k. The block is being pulled out to the right a distance x=x_0 of equilibrium and released at t = 0.
At time t_1, corresponding to \omega...
Homework Statement
The velocity of an object in simple harmonic motion is given by v(t)= -(4.04m/s)sin(21.0t + 1.00π), where t is in seconds. What is the first time after t=0.00 s at which the velocity is -0.149m/s?
Homework Equations
N/A
The Attempt at a Solution
I thought this was...
Given a general solution to the fixed-end two-mass coupled harmonic oscillator(http://teacher.pas.rochester.edu/PHY235/LectureNotes/Chapter12/Chapter12.pdf), is there a set of initial conditions for position, velocity, the 3 spring constants, and 2 masses such that a transition from random phase...
1. when wave is destructive interference ,where is the energy? for example, two plane wave have opposite phase ,they will destructive interference completely,but where is the energy? in antireflection film, the reflection wave is disappear!why? where is the energy? where is the wave?
2.in what...
Hello,
I want to include kinetic friction into the harmonic oscillator.
A small blocks is attached to a horiontal spring on a table.
Because there is kinetic friction there are two forces on the blok that we need to describe the oscillation.
First, the force that the spring exerts and second...
Trust me this is not homework... My last two questions were removed cause they looked like homework... I understand its the forum policy... From now on I will post the 'seemingly homework' on the homework sections...
Suppose,there's a rod of mass m1 hanging from a point... And a mass m2 is...
Homework Statement
A massive object of m = 5.2 kg oscillates with simple harmonic motion. Its position as a function of time varies according to the equation x(t) = 1.6sin(∏t/1.6 + ∏/6).
a. What is the position, velocity and acceleration of the object at t = 0s?
b. What is the kinetic energy...
Homework Statement
A guitar string vibrates at a frequency of 440 Hz. A point at its center moves in SHM with an amplitude of 3.0 mm and a phase angle of zero.
a. Write an equation for the position of the center of the string as a function of time.
b. What are maximum values of the magnitude...
Homework Statement
The amplitude of simple harmonic oscillation is A = 10 cm. Find a displacement x when K = 1/6 U. Here K is a kinetic energy and U is a potential energy.
Homework Equations
KE = \frac{1}{2} k A^2
U = \frac{1}{2} k x^2
The Attempt at a Solution
I'm not sure the...
Homework Statement
A cork with a density \rho0 in the form of a cube of side length l floats on water with a density of \rhow. The pressure in water depends on depth h from the surface as P=\rhow *g*h.
A. Find the equilibrium depth of the bottom surface of the cube (how much length is below...
Homework Statement
A 100g mass is suspended on a rubber band with a k coefficient of 2.74 N/m. The original amplitude of the oscillations is 5cm and after 100 oscillations, the maximum speed of the weight is 0.13 m/s. Find the damping coefficient y.
Homework Equations
d2x/dt2 + γdx/dt...
Hello
I am trying to figure out this following question:
A metronome consists of two point masses m1 and m2 on the ends of a massless rod of length
l. The top mass is m2, which is smaller than m1. The rod pivots about a point at a distance
d from m1. Use Lagrange's method to nd the...
Homework Statement
A block of mass 5 kg is attached to a spring of 2000N/m and compressed a distance of 0.6m. The spring is then released and oscillates.
a. what are the period, frequency, and angular frequency
b. what is the energy in this system
c. what is the maximum velocity...
Homework Statement
"A mass stands on a platform which executes simple harmonic oscillation in a vertical direction at a frequency of 5 Hz. Show that the mass loses contact with the platform when the displacement exceeds 10^-2 m."
Homework Equations
x(t) = a cos(wt - phi)
frequecy =...
Hi all
I'm not sure if this question is better suited for the EE thread or diff eq, but I'm trying to understand what the neper frequency, \alpha, signifies. I know it's supposed to be the damping factor and that its units are rad/second, but I'm not sure what that implies. It would seem to...
Homework Statement
1. A man's superelastic suspenders catch on a fence post, he flies back and forth, oscillating with an amplitude A. What distance does he movee in one period ? What is his displacement over 1 period
Homework Equations
x=Acos(wt)
The Attempt at a Solution
I...
I am not sure that I understand what damped harmonic oscillation is different from simple harmonic oscillation, can someone please explain that to me? I read wikipedia and still doesn't get it...
Homework Statement
A simple pendulum of mass m and lenghth l is suspended in a car traveling with a constant speed v around a circle of radius r.If the pendulum undergoes small oscillations about its equilibrium position, what will be the frequency of the oscillations??
Homework Equations...
Homework Statement
A model to describe the vibrations of atoms in a solid is to assume that the atoms are isotropic harmonic oscillators and that the vibrations are independent of the vibrations of the other atoms. We use this model to describe the entropy and heat capacity of Bohrium (B). The...