Simple Harmonic Motion

Simple Harmonic Motion (SHM) is the most fundamental oscillatory motion in physics.
A particle executes SHM when the restoring force is directly proportional to displacement from the mean position and directed towards it: $F = -kx$, where $k$ is the force constant.
This gives acceleration $a = -\omega^2 x$, where $\omega = \sqrt{k/m}$ is the angular frequency.

The general equation of SHM is $x(t) = A\sin(\omega t + \phi)$ or equivalently $x(t) = A\cos(\omega t + \phi')$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the initial phase.
The velocity $v = A\omega\cos(\omega t + \phi) = \omega\sqrt{A^2 - x^2}$ and acceleration $a = -A\omega^2\sin(\omega t + \phi) = -\omega^2 x$.

The time period $T = 2\pi/\omega = 2\pi\sqrt{m/k}$ and frequency $f = 1/T = \omega/(2\pi)$.
For a simple pendulum of length $l$: $T = 2\pi\sqrt{l/g}$ (valid for small angles $\theta < 15°$).
For a spring-mass system: $T = 2\pi\sqrt{m/k}$.

Spring Combinations:

• Series: $1/k_{\text{eff}} = 1/k_1 + 1/k_2$, so $k_{\text{eff}} = k_1k_2/(k_1+k_2)$. Time period increases.
• Parallel: $k_{\text{eff}} = k_1 + k_2$. Time period decreases.
• A spring cut into ratio $m:n$ gives spring constants in inverse ratio: $k_1 = k(m+n)/m$ and $k_2 = k(m+n)/n$.

Energy in SHM:

• Kinetic energy: $KE = \frac{1}{2}m\omega^2(A^2 - x^2)$
• Potential energy: $PE = \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}kx^2$
• Total energy: $E = \frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2$ (constant, independent of position)
• KE and PE oscillate with frequency $2\omega$ (double the SHM frequency), and their average values are each $E/2$.

Superposition of SHMs: When two SHMs of same frequency act along the same direction: $x_1 = A_1\sin(\omega t)$ and $x_2 = A_2\sin(\omega t + \phi)$, the resultant is SHM with amplitude $A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi}$.
When $\phi = 0$: $A_R = A_1 + A_2$ (constructive).
When $\phi = \pi$: $A_R = |A_1 - A_2|$ (destructive).

Damped Oscillations: In real systems, friction reduces amplitude exponentially: $x = Ae^{-bt/(2m)}\sin(\omega' t + \phi)$, where $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$ and $b$ is the damping coefficient. Energy decays as $E \propto e^{-bt/m}$.

Forced Oscillations and Resonance: When an external periodic force $F = F_0\sin(\omega_d t)$ drives a damped oscillator, the steady-state amplitude peaks when $\omega_d \approx \omega_0$ (resonance). At resonance, energy transfer from driver to oscillator is maximum. Sharpness of resonance depends inversely on damping.

The key problem-solving concept is recognizing that SHM is completely characterized by $\omega$, $A$, and $\phi$ — once these three are determined from initial conditions, every other quantity (velocity, acceleration, energy, time period) follows directly.