8.11:
Energy Diagrams – I
An energy diagram is a plot of potential energy as a function of the position of an object when a conservative force is acting upon it. Because of the net conservative forces on the object, the total mechanical energy of the system is constant.
Consider a skater at a rest point A on a frictionless parabolic ramp of height y. The gravitational potential energy of the skater, thus, is equal to mgy, while its kinetic energy is zero.
As the skater moves down, its potential energy decreases as x2.
At the bottom of the ramp, the potential energy of the skater becomes zero, while its kinetic energy reaches maximum.
When the skater moves towards point B, the potential energy reaches a maximum value with the kinetic energy approaching zero.
Further, since the ramp is frictionless, the skater performs an oscillatory movement at the bottom of the ramp. Thus, the force acting on the skater is a negative derivative of the potential energy at that given instant.
8.11:
Energy Diagrams – I
The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the interpretation of the dynamics of the system becomes even simpler.
Take the example of a skater on a parabolic ramp. The potential energy at different points along the ramp will be proportional to the height of the ramp, which varies quadratically with the horizontal position on the ramp. As the skater moves down the ramp from the highest position, their potential energy converts to kinetic energy. The net force acting on the skater is gravitational force; therefore, the total mechanical energy of the skater is constant and conserved. At any given position on the ramp, the sum of the potential energy and the kinetic energy will be the total mechanical energy of the skater.
The skater oscillates between two points, which are symmetric about the bottom of the ramp. At these two positions, the potential energies are equal, and this is the total mechanical energy of the skater, with the kinetic energy being zero. These positions are called ‘returning points’ because the skater will always move towards the bottom of the ramp from these positions. The bottom of the ramp is a position where the potential energy is zero, and the kinetic energy is equal to the total mechanical energy of the skater. This position is called the equilibrium position.
This text is adapted from Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability.