Tag Info

New answers tagged

2

A state is thermodynamically stable when its Gibbs free energy is at a minimum. $$G=U-TS+PV$$ Holding all variables but $P$ and $V$ fixed, it means that: $$dG=dP V+PdV=0 \implies {dP\over dV}=-{P\over V}$$ Since neither of $P$ nor $V$ could be negative ${dP\over dV}$ must be negative.


2

The other answers here are in the spirit of what you can do, but allow me to elaborate a little more. To understand if the trajectory of the movement under a potential $V $ is stable or not you have to understand what this stability means. The most simple example is the harmonic oscillation- $V=-{1 \over 2}kx^2 $.In Newtonian mechanics, for a point of ...


0

Once you have the value $$dV_{eff}\over dr$$ This is the minus of the effective force i.e.: $$F_{eff}=-\frac{dV_{eff}}{ dr}$$ If we differentiate this again we get: $$\frac{dF_{eff}}{dr}=-\frac{d^2V_{eff}}{ dr^2}$$ If this is negative then at any small permutation around that radius will feel a force back to that radius and thus it is a stable orbit. If it ...


0

One method for stability anayisis in a potential is to consider $r$ and $r+\epsilon$ where $\epsilon$ is a small number. If you can solve the dynamics analytically or run a numerical computer simulartion for your object with both $r$ and $r+\epsilon$ and observe with time if $\epsilon$ gets smaller or larger that will give you a clue as to whether the orbit ...


3

First of all you can directly write $p$ as a function of $V$ without solving a quadratic equation: $$ p = \frac{N k_B T}{V- N b } - a \frac{N}{V} $$ Then you take a look at the interesting points $$\left(\frac{\partial p}{\partial V}\right)_{T,N} = 0 $$ The solutions to this equation tell you where the bulk modulus (or its inverse, the compressibility) ...


0

here's a short simple answer: the Hamiltonian of the pencil can be approximated by an inverted harmonic oscillator near the equilibrium (downward parabola) . It's an easy exercise to solve.


0

Instability occurring when a heavy fluid is superposed on a lighter one was first studied by Lord Rayleigh in 1883. The nature of this instability does not change when the problem is posed as a lighter fluid accelerating against a heavier one. G I Taylor first investigated it in this latter sense. The second paper in this reply will answer your question in ...



Top 50 recent answers are included