Spring Potential Energy
U = ½kx² — stretch a spring and see the parabolic U–x curve with live reading.
Key Notes
Spring obeys Hooke's law: F = −kx (restoring force, opposite to displacement).
Elastic potential energy stored in a stretched/compressed spring: U = ½kx², where x is the displacement from natural length.
Energy stored is ALWAYS positive (x² is positive).
Conservation of energy: ½mv² + ½kx² = constant (for ideal spring-mass system, no friction).
At maximum extension/compression: KE = 0, PE = max. At natural length (passing through): KE = max, PE = 0.
Hooke's law fails for very large extensions — spring may deform plastically or break.
Springs in series: 1/k_eq = 1/k₁ + 1/k₂ (softer combination). Parallel: k_eq = k₁ + k₂ (stiffer).
Used in: oscillators, shock absorbers, vehicle suspensions, accelerometers, energy storage.
Formulas
Spring force
Negative sign: force opposes displacement (restoring).
Spring PE
Quadratic in displacement; always positive.
Energy conservation (spring-mass)
Total mechanical energy is conserved in absence of friction.
Springs (series / parallel)
Opposite of resistors (in series, springs are softer).
Important Points
U_spring is QUADRATIC in displacement — small extensions store little energy; large extensions store much.
Spring force and displacement always opposite ⇒ work done by spring = −½kx² (negative when stretching, positive when releasing).
Maximum speed of attached mass: v_max = ωx₀ = √(k/m)·x₀, where x₀ = amplitude.
Springs in series store more energy for the same displacement (1/k smaller). Parallel store less.
Hooke's law fails beyond elastic limit — energy stored differently for plastic deformation.
Common mistake: using U = ½kx (linear) instead of ½kx² (quadratic).
Spring Potential Energy 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.