Summary

• Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement and opposite in direction: $F = -kx$. The equation of motion is $\frac{d^2x}{dt^2} + \omega^2 x = 0$, where $\omega = \sqrt{k/m}$ is the angular frequency.

• Displacement in SHM is sinusoidal: $x(t) = A \sin(\omega t + \phi)$ or $x(t) = A \cos(\omega t + \phi')$, where $A$ is amplitude and $\phi$ is the initial phase.

• Velocity and acceleration in SHM: $v(t) = \omega A \cos(\omega t + \phi)$, $a(t) = -\omega^2 A \sin(\omega t + \phi) = -\omega^2 x$. Maximum speed is $v_{\text{max}} = \omega A$; maximum acceleration is $a_{\text{max}} = \omega^2 A$.

• Time period and frequency: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$, $\nu = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$.

• Energy in SHM is conserved: kinetic energy $K = \frac{1}{2} m v^2 = \frac{1}{2} k (A^2 - x^2)$, potential energy $U = \frac{1}{2} k x^2$, total energy $E = \frac{1}{2} k A^2$. Energy oscillates between $K$ and $U$ with twice the frequency of displacement.

• Simple pendulum: For small angular displacements ($\theta \lesssim 4^\circ$), it executes SHM with $T = 2\pi \sqrt{\frac{L}{g}}$, independent of mass and amplitude.

• Spring-mass system: A block of mass $m$ attached to a spring of force constant $k$ executes SHM with $T = 2\pi \sqrt{m/k}$, independent of the amplitude of oscillation.

Physical quantities used in this chapter -- symbol, dimensions, SI unit, and remarks for period, frequency, angular frequency, phase constant, and force constant

| Physical quantity | Symbol | Dimensions | Unit | Remarks |

| --- | --- | --- | --- | --- | …