Derivation of Energy in SHM




1. Potential Energy:
In Simple Harmonic Motion (SHM), the potential energy of a mass attached to a spring can be expressed as U = 1/2 kx^2, where k is the spring constant and x is the displacement from the equilibrium position. This potential energy is stored in the spring when it is stretched or compressed.

2. Kinetic Energy:
The kinetic energy of the mass in SHM can be given by K = 1/2 mv^2, where m is the mass and v is the velocity. As the mass oscillates back and forth, its kinetic energy changes continuously.

3. Total Mechanical Energy:
The total mechanical energy of the system in SHM is the sum of the potential and kinetic energies, E = U + K. This total energy remains constant throughout the motion if there are no external forces acting on the system.

4. Derivation of Energy Equation:
By substituting the expressions for potential and kinetic energies into the total mechanical energy equation, we get:
E = 1/2 kx^2 + 1/2 mv^2

5. Conservation of Energy:
In SHM, the total mechanical energy E remains constant since there are no non-conservative forces (like friction) present. This conservation of energy principle helps in understanding the behavior of the system over time.

6. Energy Exchange:
As the mass oscillates, the potential energy is converted to kinetic energy and vice versa. At the extreme points of motion (maximum displacement), all the energy is in the form of potential energy, while at the equilibrium position, all the energy is in the form of kinetic energy.

By understanding the derivation of energy in Simple Harmonic Motion, we can analyze the behavior of oscillating systems and predict their motion accurately.