1.Define Simple Harmonic Motion (SHM) and write its differential equation.
Sol: Motion in which restoring force is directly proportional to displacement towards mean position: $\frac{d^2x}{dt^2} + \omega^2 x = 0$.
2.State time period formula for a Simple Pendulum.
Sol: $T = 2\pi \sqrt{\frac{L}{g}}$.
3.Write expressions for total mechanical energy in SHM.
Sol: $E = K + U = \frac{1}{2} m \omega^2 A^2 = \text{Constant}$.
4.What is resonance in forced oscillations?
Sol: Phenomenon when driving frequency matches natural frequency of oscillator ($\omega = \omega_0$), leading to maximum amplitude.
5.A particle executes SHM with equation $x = 5 \sin(10t + \pi/4)\text{ cm}$. Find amplitude, frequency, and initial phase.
Sol: Amplitude $A = 5\text{ cm}$, $\omega = 10 \implies f = \frac{10}{2\pi} = \frac{5}{\pi} \approx 1.59 \text{ Hz}$, Initial phase $\phi = \pi/4 = 45^\circ$.
6.At what displacement from mean position is kinetic energy equal to potential energy in SHM?
Sol: $\frac{1}{2} m \omega^2 (A^2 - x^2) = \frac{1}{2} m \omega^2 x^2 \implies A^2 - x^2 = x^2 \implies 2x^2 = A^2 \implies x = \pm \frac{A}{\sqrt{2}}$.
7.Find length of a seconds pendulum ($T = 2\text{ s}$) at a location where $g = 9.8 \text{ m/s}^2$.
Sol: $T = 2\pi\sqrt{L/g} \implies 2 = 2\pi\sqrt{L/9.8} \implies L = \frac{9.8}{\pi^2} \approx 0.993 \text{ m} \approx 1 \text{ meter}$.
8.A body of mass $0.5 \text{ kg}$ suspended from a spring executes SHM with $T = 0.5 \text{ s}$. Find spring constant $k$.
Sol: $T = 2\pi\sqrt{m/k} \implies 0.5 = 2\pi\sqrt{0.5/k} \implies k = \frac{4\pi^2 (0.5)}{(0.5)^2} = 8\pi^2 \approx 78.96 \text{ N/m}$.
