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8.4 Potential energy diagrams and stability  (Page 2/7)

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You can read off the same type of information from the potential energy diagram in this case, as in the case for the body in vertical free fall, but since the spring potential energy describes a variable force, you can learn more from this graph. As for the object in vertical free fall, you can deduce the physically allowable range of motion and the maximum values of distance and speed, from the limits on the kinetic energy, 0 ≤ K ≤ E . Therefore, K = 0 and U = E at a turning point    , of which there are two for the elastic spring potential energy,

x max = ± 2 E / k .

The glider’s motion is confined to the region between the turning points, − x max ≤ x ≤ x max . This is true for any (positive) value of E because the potential energy is unbounded with respect to x . For this reason, as well as the shape of the potential energy curve, U ( x ) is called an infinite potential well. At the bottom of the potential well, x = 0 , U = 0 and the kinetic energy is a maximum, K = E , so v max = ± 2 E / m .

However, from the slope of this potential energy curve, you can also deduce information about the force on the glider and its acceleration. We saw earlier that the negative of the slope of the potential energy is the spring force, which in this case is also the net force, and thus is proportional to the acceleration. When x = 0 , the slope, the force, and the acceleration are all zero, so this is an equilibrium point    . The negative of the slope, on either side of the equilibrium point, gives a force pointing back to the equilibrium point, F = ± k x , so the equilibrium is termed stable and the force is called a restoring force. This implies that U ( x ) has a relative minimum there. If the force on either side of an equilibrium point has a direction opposite from that direction of position change, the equilibrium is termed unstable, and this implies that U ( x ) has a relative maximum there.

Quartic and quadratic potential energy diagram

The potential energy for a particle undergoing one-dimensional motion along the x -axis is U ( x ) = 2 ( x 4 − x 2 ) , where U is in joules and x is in meters. The particle is not subject to any non-conservative forces and its mechanical energy is constant at E = −0.25 J . (a) Is the motion of the particle confined to any regions on the x -axis, and if so, what are they? (b) Are there any equilibrium points, and if so, where are they and are they stable or unstable?

Strategy

First, we need to graph the potential energy as a function of x . The function is zero at the origin, becomes negative as x increases in the positive or negative directions ( x 2 is larger than x 4 for x < 1 ), and then becomes positive at sufficiently large | x | . Your graph should look like a double potential well, with the zeros determined by solving the equation U ( x ) = 0 , and the extremes determined by examining the first and second derivatives of U ( x ), as shown in [link] .

The potential energy graph U in units of Joules as a function of x in units of meters for a one-dimensional, quartic and quadratic potential energy, is shown with various quantities indicated. The horizontal scale runs from –1.2 to 1.2, labeled at 0.5 m intervals and with grid lines every 0.1 m. The vertical scale runs from –0.55 to +0.55, labeled at 0.1 J intervals with grid lines every 0.05 J. The function U of x equals 2 times the quantity x to the fourth minus x squared. This function goes to infinity at plus and minus infinite x, is zero at x equal to zero and has a minimum value of –0.5 J at x approximately equal to –0.7 m and +0.7 m. The minimum at positive x is labeled as point Q and the minimum at negative x is labeled as point minus Q. The U of x graph crosses U=0, the x axis, in two locations, at x=-1 and x=+1. The total energy E is equal to –0.25 J and is shown as a horizontal line at that value. It intersects the U of x graph at four locations, described from left to right. The leftmost point is at an x value between –0.95 and –0.9 and is labeled as point minus R. The next location at which U=-0.25 is at an x value between –0.4 and –0.35 and is labeled as point minus P. The next location at which U=-0.25 is at an x value between 0.35 and 0.4 and is labeled as point P. The rightmost location at which U=-0.25 is at an x value between 0.9 and 0.95 and is labeled as point R.
The potential energy graph for a one-dimensional, quartic and quadratic potential energy, with various quantities indicated.

You can find the values of (a) the allowed regions along the x -axis, for the given value of the mechanical energy, from the condition that the kinetic energy can’t be negative, and (b) the equilibrium points and their stability from the properties of the force (stable for a relative minimum and unstable for a relative maximum of potential energy).
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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