29
votes
Does Hooke's Law apply to all springs?
There are springs explicitly designed to deviate from Hooke's law in a repeatable and predictable manner.
(image credit)
In this example, this is done without using the possible intrinsic non-...
- 6,528
17
votes
Does Hooke's Law apply to all springs?
Hooke's law is valid for all springs when they are not "overstretched" (except possibly one designed specifically not to obey it, more on that in a bit). This also applies to other systems ...
- 8,711
14
votes
Does Hooke's Law apply to all springs?
Hooke's law applies only to springy things where the deflection response is linear, as in the case for small deflections of elastic solids.
It does not apply for nonlinear springs like that ...
- 82.8k
2
votes
Does increasing the mass in a SHM decrease the max velocity of the mass?
$$v_{\text{max}} = ω A$$
For a mass-spring oscillator, maximum velocity equals angular frequency
times amplitude. If the system is horizontal, then the equilibrium position will not change, but the ...
- 910
2
votes
Free energy of a one-dimensional harmonic oscillator
I believe that you are mixing three distinct concepts: the energy of a microstate (or configurational energy), the Helmholtz free energy, and the average (or internal) energy.
The basic idea of the ...
- 765
2
votes
Accepted
Finite quantum harmonic oscillator and existence of a ground state
The Hamiltonian of your problem is given by $$\begin{align} H &= \frac{P^2}{2m}+ V_0\left(\frac{X}{a} +1\right)\left(\frac{X}{a}-1\right) \theta(X+a) \theta(a-X) \\ &=\frac{P^2}{2m}+\frac{V_0}{...
- 2,757
2
votes
Combining two simple harmonic motion in perpendicular directions
This is an ellipse. Suppose for simplicity that the phase shift is $\pi/2$:
$$
x(t)=A\cos(\omega t + \phi),\\
y(t)=B\sin(\omega t +\phi),
$$
then
$$
\frac{x^2}{A^2}+\frac{y^2}{B^2}=1,
$$
which is an ...
- 52.3k
2
votes
Procedure to cut an Harmonic oscillator to two first level to obtain a qubit
The problem is with your truncation arrows. Truncation is not a functor.
Set $m=1, ~\omega=1$ and $\hbar=1$ w.l.o.g., to save yourself confusing elaboration. You may reinsert your nondimensionalized ...
- 54.3k
2
votes
Energy conservation in a driven harmonic oscillator
Balance of forces is not equivalent to energy conservation, and the driven oscillator is a good example of this.
Newton's second law, which is what I assume you used to find this equation of motion, ...
- 181
1
vote
Accepted
Energy conservation in a driven harmonic oscillator
If you have two harmonic quantites $x,y$:
$$
x=\Re(Xe^{i\omega t})\\
y=\Re(Ye^{i\omega t})\\
$$
then the average over a period of the product is given by:
$$
\langle xy\rangle_T=\frac{1}{2}\Re (XY^*) =...
- 5,133
1
vote
Why are the microstates of a harmonic oscillator considered equally probable, when the particle spends more time at locations of zero momentum?
You are missing the fact that there are more low-momentum states than high-momentum states. Therefore you are still more likely to observe a low momentum, although all states are equally probable. It’...
- 5,733
1
vote
Accepted
Schrödinger-Propagator for combined linear and harmonic potential
You can rewrite that as
$$V(x) = \frac{1}{2}m\omega^2x^2 + u_1x = \left(\sqrt\frac{m\omega^2}{2}x+\frac{u_1}{2}\sqrt\frac{2}{m\omega^2}\right)^2 - \frac{u_1}{2m\omega^2}.$$
Then make a coordinate ...
- 854
1
vote
If two pendulums connected by a spring both following Simply Harmonic Motion - why do they need the same time dependence?
Personally, I don't think this is intuitive at all. Every linear system has what is called normal modes$^\dagger$, solutions to the system where every component oscillates with the same frequency. A ...
1
vote
If two pendulums connected by a spring both following Simply Harmonic Motion - why do they need the same time dependence?
If you interchange the two pendulums, you have the exact same system, so they better have the same time dependence.
Otherwise, it depends on which way you're looking at: from above the page, or from ...
- 28.3k
1
vote
Occurrence of *critical* damping
To cite the comments:
Me: One could argue that critically damped oscillators are everywhere... we just don't pay attention to them, because they do not oscillate. E.g., a chair or a car suspension.
...
- 52.3k
Only top scored, non community-wiki answers of a minimum length are eligible