Simple Harmonic MotionJEE Main

Concept

Simple Harmonic Motion is oscillation where the restoring force is proportional to displacement and directed toward equilibrium: $F = -kx$. The result is sinusoidal motion $x(t) = A\cos(\omega t + \phi)$ with a single, amplitude-independent frequency.

Key formula

Derivation

Newton's second law with $F = -kx$ gives $m\ddot{x} = -kx$, i.e. $\ddot{x} = -\omega^2 x$ with $\omega^2 = k/m$.

The general solution is $x = A\cos(\omega t + \phi)$. Differentiating: $v = -A\omega\sin(\omega t+\phi)$ so $v_{max} = A\omega$, and $a = -A\omega^2\cos(\omega t+\phi)$ so $a_{max} = A\omega^2$.

Total mechanical energy is constant: $E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m\omega^2 A^2$, continuously exchanged between kinetic and potential.

Scenarios to explore

• Simple Harmonic Motion — Oscillating mass-spring: displacement, velocity, energy.

Real-world applications

• Mass-spring systems and the timing of mechanical clocks.
• Small-angle pendulum motion.
• LC electrical oscillations — the same differential equation.

JEE exam tips

• Velocity is maximum at the mean position; acceleration is maximum at the extremes.
• Use energy: $v = \omega\sqrt{A^2 - x^2}$ relates speed to displacement directly.

Common mistakes

• Thinking the period depends on amplitude — for ideal SHM it does not.
• Confusing $v_{max} = A\omega$ (at equilibrium) with $a_{max} = A\omega^2$ (at the extremes).
• Forgetting that velocity and displacement are 90° out of phase.

Exam traps to avoid

• A graph of $a$ vs $x$ for SHM is a straight line with slope $-\omega^2$.
• Phase difference matters when superposing two SHMs of the same frequency.