What is the energy conservation in oscillatory motion?

In oscillatory motion, energy is conserved due to the periodic nature of the motion.

Oscillatory motion refers to the back-and-forth motion of an object about a fixed point. Examples include the motion of a pendulum, a mass on a spring, or a vibrating guitar string. In all of these cases, the motion is periodic, meaning it repeats itself over time.

Due to the periodic nature of oscillatory motion, the total energy of the system is conserved. This means that the sum of the kinetic energy (KE) and potential energy (PE) remains constant throughout the motion. Mathematically, we can express this as:

KE + PE = constant

As the object oscillates, its kinetic energy and potential energy will vary, but their sum will always be the same. For example, in the case of a mass on a spring, the kinetic energy is at a maximum when the mass is at its equilibrium position (where the spring is neither stretched nor compressed), and the potential energy is at a maximum when the spring is at its maximum compression or extension. However, the sum of the two energies remains constant.

This conservation of energy has important implications for the motion of the object. For example, it means that the amplitude (maximum displacement) of the motion will remain constant if there is no external force acting on the system. It also means that the frequency of the motion (the number of oscillations per unit time) is determined solely by the properties of the system (such as the mass and spring constant in the case of a mass on a spring), and is independent of the initial conditions of the motion.