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# Chapter 14: Oscillation ## 14.1 Simple Harmonic Motion ### Definition Simple Harmonic Motion (SHM) is a periodic motion where the restoring force is proportional to the displacement and acts in the opposite direction. ### Equation $F = -kx$ Where: * $F$ is the restoring force * $k$ is th...

# Chapter 14: Oscillation ## 14.1 Simple Harmonic Motion ### Definition Simple Harmonic Motion (SHM) is a periodic motion where the restoring force is proportional to the displacement and acts in the opposite direction. ### Equation $F = -kx$ Where: * $F$ is the restoring force * $k$ is the spring constant * $x$ is the displacement from equilibrium ### Acceleration $a = \frac{F}{m} = -\frac{k}{m}x$ Where: * $a$ is the acceleration * $m$ is the mass ### Angular Frequency $\omega = \sqrt{\frac{k}{m}}$ ### Frequency $f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ ### Period $T = \frac{1}{f} = 2\pi\sqrt{\frac{m}{k}}$ ### Displacement $x(t) = A\cos(\omega t + \phi)$ Where: * $A$ is the amplitude * $\omega$ is the angular frequency * $t$ is the time * $\phi$ is the phase constant ### Velocity $v(t) = -A\omega\sin(\omega t + \phi)$ ### Acceleration $a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2x(t)$ ### Energy in SHM #### Potential Energy $U = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega t + \phi)$ #### Kinetic Energy $K = \frac{1}{2}mv^2 = \frac{1}{2}mA^2\omega^2\sin^2(\omega t + \phi)$ #### Total Energy $E = U + K = \frac{1}{2}kA^2 = \frac{1}{2}mA^2\omega^2$ ## 14.2 The Simple Pendulum ### Restoring Force $F = -mg\sin\theta$ Where: * $m$ is the mass * $g$ is the acceleration due to gravity * $\theta$ is the angle from the vertical ### Small Angle Approximation For small angles, $\sin\theta \approx \theta$ ### Angular Frequency $\omega = \sqrt{\frac{g}{L}}$ Where: * $L$ is the length of the pendulum ### Period $T = 2\pi\sqrt{\frac{L}{g}}$ ## 14.3 Damped Oscillations ### Damping Force $F_d = -bv$ Where: * $b$ is the damping coefficient * $v$ is the velocity ### Equation of Motion $m\frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt}$ ### Underdamped Oscillates with decreasing amplitude ### Critically Damped Returns to equilibrium as fast as possible without oscillating ### Overdamped Returns to equilibrium slowly without oscillating ## 14.4 Forced Oscillations ### Driving Force $F(t) = F_0\cos(\omega_d t)$ Where: * $F_0$ is the amplitude of the driving force * $\omega_d$ is the driving frequency ### Resonance Occurs when the driving frequency is close to the natural frequency of the system.