8.12:
Energy Diagrams – II
Consider a skater skating on a frictionless parabolic ramp, the force acting on the skater is given by the negative slope of the potential energy curve. At the bottom of the ramp, the force acting on the skater is zero.
Consider an imaginary potential energy diagram; the net force acting on the skater will be zero at point x1 and similarly at point x2.
Any displacement away from point x1 will result in a restoring force on the skater, directed towards the point x1.
Whereas, any displacement of the skater at point x2 will result in a force, directed away from the point x2.
Any minimum point in the potential energy curve is a stable equilibrium point, whereas any maximum is an unstable equilibrium point.
If the total energy of the skater is E2, then the skater will be able to escape and go beyond point x2. But if the total energy is E1, then the skater will be trapped between positions x0 and x2.
8.12:
Energy Diagrams – II
Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the system.
The point in the energy diagram at which the system’s potential energy is the lowest is known as the local minima. The system tends to stay in this position indefinitely unless acted upon by a net force. The slope of the potential energy diagram at the local minima is zero, indicating that zero net force is acting on the system. The slope is positive on either side of the minima, which suggests that the net force acting upon the system is a restoring one. Similarly, the net force acting on the system at the local maxima is also zero, but this is an unstable equilibrium point. Any movement away from this point results in a force that is acting away from the point.
There will always be a neutral equilibrium point characterized by the constant potential energy regime of the potential energy diagram, where the net force on the system will be zero. Any movement on either side of the neutral equilibrium position will result in a net force on the system, neither restoring nor disrupting.
This text is adapted from Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability.