Energy in Simple Harmonic Motion (SHM)
IIT‑JEE (Main + Advanced) – Physics – “Energy in SHM”
1. Simple Explanation
In a simple harmonic oscillator (a mass on a spring, a small‑angle pendulum, etc.) the restoring force is proportional to the displacement:
[ F = -k,x \quad\text{or}\quad F = -m\omega^{2}x . ]
Because the force is conservative, the mechanical energy (kinetic + potential) is conserved.
During the oscillation the energy oscillates between kinetic energy (max at the equilibrium position) and potential energy (max at the turning points).
The total energy is a constant that depends only on the amplitude of motion, not on the time or phase.
2. Key Points
| Concept | Formula | Meaning |
|---|---|---|
| Potential Energy (spring) | (U = \dfrac12 kx^{2}) | Energy stored in the stretched/compressed spring. |
| Potential Energy (pendulum, small‑angle) | (U = \dfrac12 m g L,\theta^{2}) | Gravitational PE with (\theta) the angular displacement. |
| Kinetic Energy | (K = \dfrac12 m v^{2}) | (v = \dot{x} = \omega A\sin(\omega t+\phi)). |
| Total Mechanical Energy | (E = K + U = \dfrac12 k A^{2} = \dfrac12 m \omega^{2}A^{2}) | Constant for an undamped oscillator. |
| **Amplitude |