This is equivalent to having the torques of the individual parts balanced about the pivot point, in this case the hand. The cgs of the arms, legs, head, and torso are labeled with smaller type. A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement.
For example, a marble at the bottom of a bowl will experience a restoring force when displaced from its equilibrium position. This force moves it back toward the equilibrium position. Most systems are in stable equilibrium, especially for small displacements. For another example of stable equilibrium, see the pencil in Figure 2.
Figure 2. This pencil is in the condition of equilibrium. The net force on the pencil is zero and the total torque about any pivot is zero. A system is in unstable equilibrium if, when displaced, it experiences a net force or torque in the same direction as the displacement from equilibrium.
A system in unstable equilibrium accelerates away from its equilibrium position if displaced even slightly. An obvious example is a ball resting on top of a hill. Once displaced, it accelerates away from the crest. See the next several figures for examples of unstable equilibrium. Figure 3. If the pencil is displaced slightly to the side counterclockwise , it is no longer in equilibrium. Its weight produces a clockwise torque that returns the pencil to its equilibrium position. Figure 4.
If the pencil is displaced too far, the torque caused by its weight changes direction to counterclockwise and causes the displacement to increase. Figure 5. This figure shows unstable equilibrium, although both conditions for equilibrium are satisfied. Figure 6. If the pencil is displaced even slightly, a torque is created by its weight that is in the same direction as the displacement, causing the displacement to increase.
A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position. A marble on a flat horizontal surface is an example. Combinations of these situations are possible. For example, a marble on a saddle is stable for displacements toward the front or back of the saddle and unstable for displacements to the side.
Figure 6 shows another example of neutral equilibrium. Figure 7. The cg of a sphere on a flat surface lies directly above the point of support, independent of the position on the surface. The sphere is therefore in equilibrium in any location, and if displaced, it will remain put. When we consider how far a system in stable equilibrium can be displaced before it becomes unstable, we find that some systems in stable equilibrium are more stable than others.
The pencil in Figure 2 and the person in Figure 8 a are in stable equilibrium, but become unstable for relatively small displacements to the side. The critical point is reached when the cg is no longer above the base of support. This control is a central nervous system function that is developed when we learn to hold our bodies erect as infants. This statement has a mathematically precise form known as the Hartman-Grobman Theorem. One says that these systems are locally topologically conjugate equivalent.
That is, adding nonlinear terms to a linear system at a hyperbolic equilibrium may distort but does not change qualitatively the phase portrait near the equilibrium. If at least one eigenvalue of the Jacobian matrix is zero or has a zero real part, then the equilibrium is said to be non-hyperbolic. Non-hyperbolic equilibria are not robust i. Some refer to such an equilibrium by the name of the bifurcation , e.
In practice, one often has to consider non-hyperbolic equilibria with all eigenvalues having negative or zero real parts. These equilibria are sometimes referred to as being critical. Nevertheless, a non-hyperbolic equilibrium of a one-dimensional system is stable if the function changes sign from positive to negative at the equilibrium.
It has two eigenvalues, which are either both real or complex-conjugate. A hyperbolic equilibrium can be a. Figure 3 summarizes the types of equilibria. The Jacobian matrix of a three-dimensional system has 3 eigenvalues, one of which must be real and the other two can be either both real or complex-conjugate. Depending on the types and signs of the eigenvalues, there are a few interesting cases illustrated in Figure 4. A hyperbolic equilibrium can be. Style: MLA.
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