Simple Harmonic Motion (JEE)

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1. Introduction to SHM Definition: A special type of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and acts opposite to the displacement. Condition for SHM: $F = -kx$ or $a = -\omega^2 x$. Equation of SHM: $m \frac{d^2x}{dt^2} = -kx \Rightarrow \frac{d^2x}{dt^2} + \frac{k}{m}x = 0$. Comparing with standard form $\frac{d^2x}{dt^2} + \omega^2 x = 0$, we get $\omega = \sqrt{\frac{k}{m}}$. 2. Key Parameters Angular Frequency ($\omega$): $\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T} = 2\pi f$ (rad/s) Time Period ($T$): Time taken for one complete oscillation. $T = 2\pi\sqrt{\frac{m}{k}} = \frac{1}{f}$ (s) Frequency ($f$): Number of oscillations per second. $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} = \frac{1}{T}$ (Hz) Amplitude ($A$): Maximum displacement from the mean position. 3. Displacement, Velocity, and Acceleration Displacement ($x$): General form: $x(t) = A \sin(\omega t + \phi)$ or $x(t) = A \cos(\omega t + \phi)$ If starting from mean position ($x=0$ at $t=0$): $x(t) = A \sin(\omega t)$ If starting from extreme position ($x=A$ at $t=0$): $x(t) = A \cos(\omega t)$ $\phi$ is the initial phase. Velocity ($v$): $v(t) = \frac{dx}{dt} = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2}$ Maximum velocity ($v_{max}$): $A\omega$ (at mean position, $x=0$) Minimum velocity ($v_{min}$): $0$ (at extreme positions, $x=\pm A$) Acceleration ($a$): $a(t) = \frac{dv}{dt} = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ Maximum acceleration ($a_{max}$): $A\omega^2$ (at extreme positions, $x=\pm A$) Minimum acceleration ($a_{min}$): $0$ (at mean position, $x=0$) 4. Energy in SHM Kinetic Energy ($KE$): $KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2 - x^2) = \frac{1}{2}k(A^2 - x^2)$ Maximum KE: $\frac{1}{2}m\omega^2 A^2 = \frac{1}{2}kA^2$ (at mean position) Potential Energy ($PE$): $PE = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2$ Maximum PE: $\frac{1}{2}kA^2$ (at extreme positions) Total Mechanical Energy ($E$): $E = KE + PE = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ Total energy remains constant throughout SHM. 5. Simple Pendulum For small angular displacements ($\theta Restoring Force: $F = -mg \sin\theta \approx -mg\theta = -mg \frac{x}{L}$ Effective Spring Constant: $k_{eff} = \frac{mg}{L}$ Time Period ($T$): $T = 2\pi\sqrt{\frac{L}{g}}$ Effect of Length: $T \propto \sqrt{L}$ Effect of Gravity: $T \propto \frac{1}{\sqrt{g}}$ Seconds Pendulum: $T=2s$, $L \approx 1m$ on Earth. 6. Spring-Mass System Horizontal Spring: Mass $m$ attached to a spring with constant $k$. Time Period: $T = 2\pi\sqrt{\frac{m}{k}}$ Vertical Spring: At equilibrium, $mg = kx_0$ (where $x_0$ is static elongation). If displaced from equilibrium, the net restoring force is $F = k(x_0+x) - mg = kx_0 + kx - mg = kx$. Time Period: $T = 2\pi\sqrt{\frac{m}{k}}$ (same as horizontal spring, independent of $g$) Springs in Series: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + ...$ Springs in Parallel: $k_{eq} = k_1 + k_2 + ...$ 7. Physical Pendulum Any rigid body oscillating about a fixed horizontal axis. Time Period ($T$): $T = 2\pi\sqrt{\frac{I}{mgd}}$ $I$: Moment of inertia about the axis of oscillation. $m$: Mass of the body. $g$: Acceleration due to gravity. $d$: Distance from the center of mass to the axis of oscillation. Length of equivalent simple pendulum: $L_{eq} = \frac{I}{md}$ 8. Torsional Pendulum A body suspended by a wire, oscillating due to torsional rigidity. Restoring Torque: $\tau = -C\theta$ (where $C$ is torsional constant) Time Period ($T$): $T = 2\pi\sqrt{\frac{I}{C}}$ $I$: Moment of inertia of the body about the suspension wire. 9. Damped Oscillations Occurs when a resistive force (e.g., air drag, friction) opposes the motion. Damping Force: $F_d = -bv$ (where $b$ is damping constant) Equation of Motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$ Displacement: $x(t) = A_0 e^{-(b/2m)t} \sin(\omega' t + \phi)$ Amplitude $A(t) = A_0 e^{-(b/2m)t}$ decreases exponentially. Angular frequency $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$, where $\omega_0 = \sqrt{k/m}$. Critical Damping: $b^2 = 4mk$ (returns to equilibrium fastest without oscillation). Over Damping: $b^2 > 4mk$ (returns to equilibrium slowly without oscillation). Under Damping: $b^2 10. Forced Oscillations & Resonance When an external periodic force $F(t) = F_0 \sin(\omega_d t)$ acts on an oscillating system. Equation of Motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \sin(\omega_d t)$ Steady-State Solution: $x(t) = A \sin(\omega_d t + \delta)$ Amplitude ($A$): $A = \frac{F_0}{\sqrt{m^2(\omega_d^2 - \omega_0^2)^2 + b^2\omega_d^2}}$ Resonance: Occurs when the driving frequency $\omega_d$ is close to the natural frequency $\omega_0$ of the system. For small damping, maximum amplitude occurs when $\omega_d \approx \omega_0$. At resonance, the amplitude of oscillation becomes very large. Quality Factor ($Q$): $Q = \frac{\omega_0}{\Delta\omega} = \frac{m\omega_0}{b}$ A higher $Q$ factor means sharper resonance and less damping.