1.Define Simple Harmonic Motion (SHM) and write its standard differential equation $\frac{d^2x}{dt^2} + \omega^2 x = 0$.
Sol: Motion in which restoring force is directly proportional to displacement towards mean position: $F = -k x \implies m \frac{d^2x}{dt^2} + k x = 0 \implies \frac{d^2x}{dt^2} + \omega^2 x = 0$.
2.Derive the time period of a Simple Pendulum $T = 2\pi\sqrt{\frac{L}{g}}$.
Sol: Restoring torque $\tau = -m g L \sin\theta \approx -m g L \theta = I \alpha = (m L^2) \alpha \implies \alpha = -\left(\frac{g}{L}\right) \theta \implies \omega = \sqrt{\frac{g}{L}} \implies T = 2\pi \sqrt{\frac{L}{g}}$.
3.Derive expression for Kinetic Energy, Potential Energy, and Total Energy of a particle executing SHM.
Sol: $PE = \frac{1}{2} m \omega^2 x^2$, $KE = \frac{1}{2} m \omega^2 (A^2 - x^2)$, $TE = PE + KE = \frac{1}{2} m \omega^2 A^2 = \text{Constant}$.
