Acceleration vs Time
a = -ω² x — 180° out of phase.
Key Notes
Acceleration in SHM: a(t) = dv/dt = −Aω²·cos(ωt + φ) = −ω²·x.
Maximum |a| = Aω², at the extremes (x = ±A).
ZERO at equilibrium (x = 0).
Always points TOWARD the equilibrium position (restoring nature).
a vs x: linear with negative slope (a = −ω²x). On an a-x graph: straight line through origin with slope −ω².
a is in PHASE with −x: when x is at +A, a is at maximum negative value.
Important relation: a + ω²x = 0 — the defining DE of SHM.
Used in finding force, identifying SHM from given motion equations.
Formulas
Acceleration
Second derivative of x.
Maximum acceleration
At x = ±A.
Force in SHM
Restoring force, with k = mω².
a-x relation
Straight line through origin on a-x graph; slope = −ω².
Important Points
a is MAX at extremes (|x| = A) and ZERO at equilibrium.
a always points toward x = 0 — that's why oscillation continues.
a is 180° out of phase with x (same frequency, opposite sign).
Identifying SHM: check if a = −const × x. If yes, it's SHM with ω = √(const).
a leads v by π/2 (a = max when v = 0 at extremes; a = 0 when v = max at equilibrium).
Common pitfall: drawing a-x curve as nonlinear. It's straight: a = −ω²x.
Acceleration vs Time 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.