Author Capstone Axis

Velocity & Acceleration in Simple Harmonic Motion Velocity & Acceleration in Simple Harmonic Motion 🚀 Quick Recap of SHM ➡️ Displacement: \(x(t)=A\cos(\omega t+\phi)\) Amplitude \(A\) is the farthest distance from the mean position. Angular frequency \(\omega=\dfrac{2\pi}{T}\), where \(T\) is the period. Instantaneous Velocity 🏃‍♂️ In a circle of radius \(A\) the speed is \(v=\omega A\). […]

Simple Harmonic Motion – Force Law, Motion & Energy 🚀 1. How the motion looks in time ⏱️ Displacement: \(x(t)=A\cos(\omega t+\phi)\) — a comfy cosine ride starting at amplitude \(A\). :contentReference[oaicite:0]{index=0} Speed: \(v(t)=-\omega A\sin(\omega t+\phi)\) — reaches \(\pm\,\omega A\) at the mid-points. :contentReference[oaicite:1]{index=1} Acceleration: \(a(t)=-\omega^{2}A\cos(\omega t+\phi)\) — maxes out at \(\pm\,\omega^{2}A\) right at the ends.

Energy in Simple Harmonic Motion 🌟 In simple harmonic motion (SHM), energy keeps shuffling between kinetic and potential forms while the total stays fixed—like coins moving between two pockets but never leaving your jeans! 😄 1. Kinetic Energy (K) 🚀 K comes from speed (v) and peaks at the center (x = 0). It is

Simple Pendulum 🕰️ Galileo famously timed the regular swings of a church chandelier with his own pulse and discovered a beautiful rhythm hiding in plain sight. You can feel the same wonder by tying a small stone to a ~100 cm thread and letting it swing! Each gentle to-and-fro takes about 2 s. 🎶 :contentReference[oaicite:0]{index=0}

🌊 1. Big Picture – What really is a wave? When you toss a pebble into still water, ripples race outward, yet the water itself doesn’t drift away. That traveling pattern is a wave – a disturbance that carries energy and information from one spot to another without hauling matter along for the ride. Your

Transverse & Longitudinal Waves 🎸🌊 1. Waves in a Nutshell Mechanical waves move energy from one place to another by making the medium’s particles oscillate while the material itself stays put. Think of a crowd doing “the wave” at a stadium —the cheer travels, people don’t! 🌊:contentReference[oaicite:0]{index=0} 2. Two Main Flavors 😉 2.1 Transverse Waves

Displacement Relation in a Progressive Wave 🌊 Picture a travelling wave as a moving “wiggle” that carries energy without dragging the material along. For a smooth (sinusoidal) wave moving in the +x-direction you use \(y(x,t)=a\sin\bigl(kx-\omega t+\varphi\bigr)\) :contentReference[oaicite:0]{index=0} \(y(x,t)\): displacement of the particle at position \(x\) and time \(t\). \(a\): amplitude – the largest distance a

Speed of Travelling Waves 🚀 1 • Chasing the Crest Track a point of fixed phase on the wave—say the crest—and keep its phase constant: \(k\,x – \omega\,t = \text{constant}\) 🙂 From this condition you get the speed \(v = \dfrac{\omega}{k}\). Because \(k = \dfrac{2\pi}{λ}\) and \(\omega = 2\pi\nu = \dfrac{2\pi}{T}\), the same speed appears in the more familiar

1. Principle of Superposition 🌊➕🌊 When two wave pulses travel toward each other, nothing dramatic happens—they simply glide through one another. While they overlap, every particle shifts by the straight-forward sum of the individual shifts. For two waves this reads: \( y(x,t)=y_1(x,t)+y_2(x,t) \)   (Eq. 14.25) :contentReference[oaicite:0]{index=0} If more than two waves meet, just keep adding

Superposition & Interference 🎶 The disturbance in a medium adds up algebraically. For many individual disturbances \(y_1,\,y_2,\dots,y_n\) travelling with the same speed \(v\), the combined wave is $$y(x,t)=\sum_{i=1}^{n}f_i\!\bigl(x-vt\bigr) \tag{14.26}$$ :contentReference[oaicite:0]{index=0} With just two harmonic waves of equal amplitude \(a\) and wavelength \(\lambda\), but a phase gap \(\phi\), the net displacement becomes $$y(x,t)=2a\cos\!\Bigl(\tfrac{\phi}{2}\Bigr)\,\sin(kx-\omega t+\tfrac{\phi}{2}) \tag{14.31}$$ :contentReference[oaicite:1]{index=1}