Energy in SHM

Energy is central to understanding SHM. During each oscillation, energy is continuously exchanged between kinetic and potential forms, but the total mechanical energy remains constant (in the absence of damping). This energy analysis provides deep insight into why the motion behaves as it does and gives us another powerful equation for solving problems.

---

Kinetic Energy in SHM

The kinetic energy of an oscillating object at any displacement x is:

KE = ½mv²

Using the velocity-displacement relation v² = ω²(A² − x²), we can substitute:

KE = ½mω²(A² − x²)

Key features:

• KE is maximum at the equilibrium position (x = 0): KE_max = ½mω²A²
• KE is zero at maximum displacement (x = ±A): the object is momentarily stationary
• KE varies as a function of displacement — it decreases as the object moves away from equilibrium

---

Potential Energy in SHM

The potential energy stored in the system at displacement x is:

PE = ½mω²x²

For a mass-spring system, this is elastic potential energy stored in the stretched or compressed spring (PE = ½kx², and since ω² = k/m, we get ½kx² = ½mω²x²). For a pendulum, it is gravitational potential energy.