Energy in SHM (KE ↔ PE)
E = ½kA² constant — KE/PE swap.
Key Notes
Energy in SHM oscillates between KE and PE but total stays constant.
KE: ½mv² = ½m·ω²·(A² − x²). Maximum (½mω²A²) at x = 0, zero at x = ±A.
PE: ½kx² = ½m·ω²·x². Maximum (½mω²A²) at x = ±A, zero at x = 0.
Total: E = ½kA² = ½mω²A² — depends on amplitude squared.
PE and KE oscillate at TWICE the SHM frequency (sin² and cos² have period T/2).
Time-average of KE over a period = time-average of PE = ½ × Total energy.
Doubling amplitude QUADRUPLES total energy.
Energy conservation: ½kx² + ½mv² = ½kA² always.
Formulas
Total energy
Constant. Proportional to A².
Kinetic energy
Max at x = 0, zero at x = ±A.
Potential energy
Max at extremes, zero at equilibrium.
Energy conservation
At every instant.
Time-averages over T
Each is half the total energy.
Important Points
Total energy is independent of time — KE and U swap with period T/2.
Energy is QUADRATIC in amplitude (E ∝ A²). Doubling A ⇒ quadruple E.
KE = PE when x = A/√2.
On a graph: E is a horizontal line, KE is an inverted parabola peaking at center, U is an upright parabola.
Damped oscillation: E decreases exponentially as amplitude decays.
Adding mass to a spring-mass system doesn't change E if A is fixed — but it changes ω and T.
Energy in SHM (KE ↔ PE) notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.