Physics
Simple Harmonic Motion
Simple harmonic motion is the back-and-forth oscillation when the restoring force is proportional to displacement (F = -kx). Period T = 2π·√(m/k) for a spring; T = 2π·√(L/g) for a small-angle pendulum. The position oscillates sinusoidally.
How to Approach Simple Harmonic Motion
Identify the system
Spring–mass: F = -kx, so ω = √(k/m). Pendulum (small angle): ω = √(g/L). Note ω is angular frequency in rad/s, not regular frequency f = ω/(2π).
Write the equation of motion
x(t) = A·cos(ωt + φ). A is amplitude, ω is angular frequency, φ is phase. Velocity v(t) = -Aω·sin(ωt + φ). Acceleration a(t) = -Aω²·cos(ωt + φ).
Use energy if asked
At maximum displacement: all energy is PE = ½kA². At equilibrium: all energy is KE = ½mv_max². Energy conservation links them: ½kA² = ½m·v_max² → v_max = Aω.
Frequently Asked Questions
What's resonance?+
When you drive an SHM system at its natural frequency, amplitude grows without bound (in the absence of damping). Why a singer can shatter a glass, why bridges can collapse, why MRI works.
Why does a pendulum's period not depend on mass?+
Because gravity provides both the restoring force (proportional to mass) and the inertia (proportional to mass). Mass cancels out. The period depends only on L and g.
When does the small-angle approximation fail?+
Above about 15-20°, sin(θ) ≈ θ stops being accurate enough. The period gets slightly longer, and SHM is replaced by elliptic-integral equations of motion.
Related Topics
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