Energy plot and Types of Equilibrium
Photo by Pixabay
A stable equilibrium is when the body returns to its original position.
Unstable equilibrium is when the body does not return to its initial position.
Neutral equilibrium is when the center of gravity does not change.
We know E= T + U (or V) from our previous discussion in the blog post.
Here, consider a point particle under the influence of a conservative force with potential U.
We first find the expression of velocity and then of time.
For expression of velocity from energy
For expression of time from velocity
Discussion of motion of particle examining a plot for potential
In the plot, we show the potential energy of the system with varies energies.
For any real system, K.E> 0
And so E> potential energy
Because the expression of v as obtained in figure 1, tells that velocity v is real if and only if potential is less than energy.
Turning points of motion
If the potential energy of the system becomes equal to energy E by Image1’s expression of velocity, we have a velocity equal to zero. This means the particle will come to rest at these points and then reverse its direction to its original direction. These points are called turning points of motion.
The plot goes here.
1. Here E is the energy.
2. The regions between points x1 and x4 and points x5 and x7 are not allowed in other words regions 2 and 4 are forbidden. Here the K.E. is negative so the velocity of the particle is not real.
3. All other points are allowed.
4. Region x1,x3,x5,x7 are turning points of motion.
5. Region 3 has 2 turning points and so it is a bounded system. Here the particle executes bounded motion. If there’s a one-dimensional case, bounded motion is necessarily periodic.
6. Regions 1 and 5 have each one turning point, the other point for such region is unbounded.
7. Region 1 included Origin and the particle can pass through it. However, classically we cannot always solve a problem when the particle comes through its origin it comes very close to it.
8. Regions 2 and 4 are bounded regions but are forbidden. Region 3 is the Potential well.
Stable and Unstable Equilibrium
In the plot slope of the potential function vanishes at x4, x6, x8, and x9.
Then,
dV(x)/dx = 0
Force equals potential gradient so firce is zero at those points.
When a particle is placed there with zero velocity, it remains at rest. The particle is said to be in equilibrium. Such points are called equilibrium points.
Stable and Unstable points
x4, and x8 have minimum potential values and so they are stable points.
x6, x9 has maximum potential energy values, they are unstable.
Let xe be the equilibrium point. If the displacement of the particle from the equilibrium point is small, then we can approximate the potential by series expansion x=xe as:
Subscript x=xe means that quantity is to be found out at x=xe.
Now at x=xe, dV/dx = 0
For Configuration near xe, the x square term is large. The other powers of c are small and can be ignored.
Stable Equilibrium
* If the second derivative of V w.r.t x is positive, V(x)— V(xe) is positive for all values of x implying V(x) is minimum at x.
If a particle is in a situation at xe and it experiences a slight disturbance m, it will return to xe when the disturbance is removed this is called stable equilibrium.
Unstable Equilibrium
* If second derivative of V w.r.t x is negative, V(x) < V(xe) for all values of x.
If the particle is slightly displaced from xe it will move away from the original position of equilibrium when the disturbance is removed.
Critical and Neutral Equilibrium:
* If the second derivative of V w.r.t x at x=xe is zero, the equilibrium is critical.
Further evaluation of critical equilibrium
Introduce further term in Taylor expansion.
The most important term in value if V(x) - V(xe) is (x-xe)^3
The figure sums up the general rules.
Neutral Equilibrium
If V(x) is constant throughout the whole range surrounding the under consideration Configuration, then the body can be displaced and there will be no force that tends to move the body away from the new Configuration. Thus every Configuration is one of equilibrium. This kind of equilibrium is neutral equilibrium.
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