11.12:
Stability of Equilibrium Configuration: Problem Solving
Consider a disk connected to a spring pulley system with a ball attached to its rim.
The disk is rotated and the free-body diagram of only the forces that do work is drawn.The deflections on rotations are shown with respect to coordinate axis.
To determine the equilibrium positions of the system and their stability, the sum of the elastic and gravitational potential energy functions is obtained.
The vertical deflection of the spring and the ball is determined and substituted into the potential energy function.
At equilibrium, the first derivative of the potential energy function is zero. Solving and substituting the numerical values gives the equilibrium positions in terms of rotation angle.
The stability of the equilibrium angles is determined from the second derivative of the potential energy.
If the second derivative is negative, the equilibrium is unstable. At this angle, a slight movement sets the system in rotation.
If it is positive, the equilibrium is stable, implying that, under rotation, the system comes to rest at this angle.
11.12:
Stability of Equilibrium Configuration: Problem Solving
The stability of equilibrium configurations is an important concept in physics, engineering, and other related fields. In simple terms, it refers to the tendency of an object or system to return to its equilibrium position after being disturbed. The stability of an equilibrium configuration can be analyzed by considering the potential energy function of the system and examining its behavior near the equilibrium point.
Problem-solving in the context of the stability of equilibrium configuration involves several steps. Firstly, one needs to identify the equilibrium points of the system. This can be achieved by setting the derivative of the potential energy function to zero and solving for the values of the independent variables that satisfy the equation.
Once the equilibrium points have been determined, the next step is to examine their stability. To do this, the second derivative of the potential energy function is evaluated at each equilibrium point. If the second derivative is positive, the equilibrium is stable, indicating that the system will return to the equilibrium position when disturbed. Conversely, if the second derivative is negative, the equilibrium is unstable, meaning the system will move away from the equilibrium position when disturbed.
It is important to note that a system may have multiple equilibrium points, and each must be analyzed separately to determine its stability. Furthermore, the stability of an equilibrium configuration can be affected by various factors, such as the presence of damping or external forces.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Section 11.7, Pp. 603-606.
- Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. and Self, B.P. (2016). Vector Mechanics For Engineers. Eleventh Edition, New York: McGraw-Hill Education. Pp. 601.