20 votes

Can motion be oscillatory but not periodic?

The graph of $cos^2(t)$ (blue) oscillates and is periodic in time. The graph of $cos(t^2)$ (green) oscillates but is not periodic. The time between cycles is always decreasing The answer to this ...
Stevan V. Saban's user avatar
14 votes

Can motion be oscillatory but not periodic?

The formula is ambiguous! $$ \cos( (\alpha t)^2 ) $$ oscillates but is not periodic. $$ ( \cos(\alpha t) )^2 $$ oscillates and is periodic.
Andrew Steane's user avatar
10 votes
Accepted

Can motion be oscillatory but not periodic?

If the expression is intended to be $x = a (\cos (\alpha t))^2$ then this is most definitely periodic, with period $\frac {\pi}{\alpha}$. In other words: $x(t) = x(t \pm \frac {\pi} {\alpha}) = x(t \...
gandalf61's user avatar
  • 53.1k
8 votes

Can motion be oscillatory but not periodic?

I agree with others in that the "definition" (if you can call it that) of oscillatory in your book leaves something to be desired, and damped oscillation as suggested by LolloBoldo is still ...
Jyrki Lahtonen's user avatar
6 votes

Can motion be oscillatory but not periodic?

Periodic usually means that: $$x(t+ T) = x(t)$$ Where $T$ is the period. Based on that clearly $x(t) =K\cos(\alpha t)^2$ is periodic of period $T= \pi/\alpha$. If one assume that oscillatory means ...
LolloBoldo's user avatar
  • 1,390
5 votes
Accepted

Time for pendulum to leave unstable equilibrium is logarithmic

The logarithmic divergence is correct. To make things simple, I will consider the dimensionless system: $$ \ddot \theta=\sin\theta $$ with energy: $$ H = \frac12\dot\theta^2+\cos\theta $$ This is ...
LPZ's user avatar
  • 11.5k
4 votes

Can motion be oscillatory but not periodic?

I would question the notion that the usual definition of "oscillatory" implies periodicity. Otherwise the term "chaotic oscillation" does not make sense, but indeed it is used a ...
Peter Rottengatter's user avatar
4 votes

Can motion be oscillatory but not periodic?

I would formalise oscillatory and periodic as follows: Take a function $f: \mathbb{R} \to Y$, where $\mathbb{R}$ are the time-values and $Y$ are the possible values of $f$ (most often position, ...
Jannik Pitt's user avatar
3 votes
Accepted

Directly integrating the Lagrangian for a simple harmonic oscillator

Well, if we know the classical solution $q_{\rm cl}:[t_i,t_f] \to \mathbb{R}$ (which we do for the harmonic oscillator), we can plug it into the action functional $S[q]$ and obtain the on-shell action ...
Qmechanic's user avatar
  • 202k
3 votes
Accepted

Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$

When we talk about the quantum harmonic oscillator, $a$ and $a^{\dagger}$ are usually used.($a$ may stands for "annihilation".) Note that we have not discuss second quantization here, so $a$ ...
Gordon Liu's user avatar
2 votes

Can motion be oscillatory but not periodic?

For example, Bessel functions are oscillatory but not periodic.
P-Dub's user avatar
  • 21
2 votes
Accepted

Plot of Number of Oscillations of a Pendulum

The expected number of oscillations in a period $\Delta t$ is $N = \Delta t / T$. In terms of your quantity $x = l/g (\Delta t)^2$, you have $\Delta t = \sqrt{l/gx}$. If you combine these two ...
Michael Seifert's user avatar
2 votes

Ladder operators and creation & annihilation operators - different between $a$, $b$ and $c$

Someone can tell me the different between the notation $a$ or $b$ or $c$? The name of these operators doesn't matter. The important thing are the commutation relations between these two operators (...
Thomas Fritsch's user avatar
2 votes
Accepted

Why can we ignore the work done by gravity?

So step one is don't put number in equations until the end. And, if you do put numbers in: include units. Something like $(2)(3)^2$ is meaningless, and I've done hundreds of spring problems. Step 2: ...
JEB's user avatar
  • 33.7k
2 votes
Accepted

Shape of graph of energy in S.H.M

For a simple harmonic motion we have $$x(t)=x_\max\sin(\omega t+\phi_0)$$ $$\dot{x}(t)=x_\max\omega\cos(\omega t+\phi_0)$$ Hence we get the kinetic energy $$\begin{align} E_\text{kin}&=\frac{1}{2}...
Thomas Fritsch's user avatar
1 vote

Quantum harmonic oscillator hamiltonian linkage to the parity operator

I strongly believe you "misheard" your homework problem, which, instead, should have been something like "Write the parity operator in terms of the SHO hamiltonian". The Parity ...
Cosmas Zachos's user avatar
1 vote

Directly integrating the Lagrangian for a simple harmonic oscillator

This question can be cast in a convenient way by using the initial conditions $$ x_a=x(t_a) \quad x_b=x(t_b), $$ and taking $$ x(t)=A(x_a,x_b)\cos(\omega t)+B(x_a,x_b)\sin(\omega t). $$ Then, you can ...
Jon's user avatar
  • 3,828
1 vote
Accepted

Doubt obtaining the expected value of $x^2$ of a bidimensional harmonic oscillator

You need to calculate the expectation value in the position basis: \begin{align} \langle x^2 \rangle &= \langle \phi | x^2 | \phi \rangle = \int \mathrm{d}x \int \mathrm{d}y \; \phi^*(x,y) x^2 \...
WillHallas's user avatar
1 vote
Accepted

Can we equate SHM with motion in a circle?

For circular motion in the plane the position is written as: $$\vec r(t)=r\cos(\omega t)\mathbf{\hat x}+r\sin(\omega t)\mathbf{\hat y}.$$ Taking the time derivative we get the velocity: $$\vec v(t)=-r\...
Albertus Magnus's user avatar
1 vote

Can we equate SHM with motion in a circle?

Is the following correct ? No. You are trying to equate two different motions - SHM, which is a one-dimensional motion with varying speed, and circular motion, which is a two dimensional motion at ...
gandalf61's user avatar
  • 53.1k
1 vote

What is the equation if that projection starts SHM on the $x$-axis from extreme position?

Start in polar coordinates: $$ r = A $$ $$ \theta(t) = \omega t + \theta_0 $$ where $\theta_0$ is the phase at $t=0$. Vary $\theta_0$ to create the initial position, and convert to Cartesian ...
JEB's user avatar
  • 33.7k
1 vote
Accepted

Does it make sense to talk about individual energies of interacting quantum particles?

It depends on what you are looking for. If you want to move away from mathematical rigor and more to situations in daily life where you need practical communication, then I think you will find many ...
Dr. Nate's user avatar
  • 368

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