broski
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@@ -250,9 +250,11 @@ $$
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### **How to Calculate $ n $:**
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Since $ n $ counts both nodes and antinodes:
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$$
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n = \text{total nodes} + \text{total antinodes}
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n = \text{total nodes} + \text{total antinodes}-1
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$$
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The above is true only if you cound the starting node of the fixture of the air column. You do not subtract by $1$ otherwise.
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---
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## **Both Ends Open (Pipe Open at Both Ends)**
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@@ -289,3 +291,142 @@ $$
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| **Both Ends Open** | Antinode at both ends | $L = n \frac{\lambda}{2}$ | $f_n = \frac{n v}{2L}$ | $n = \text{antinodes}$ |
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Now the formulas **remain internally consistent with a fixed definition of $ n $** across all cases.
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# Pendulum formulas
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Here are the main pendulum formulas in general physics:
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1. **Period of a simple pendulum** (small angle approximation):
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$$
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T = 2\pi \sqrt{\frac{L}{g}}
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$$
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Where:
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- $T$ = period (time for one full swing)
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- $L$ = length of the pendulum
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- $g$ = acceleration due to gravity
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2. **Frequency of a simple pendulum**:
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$$
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f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{L}}
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$$
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Where:
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- $f$ = frequency (number of oscillations per second)
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3. **Angular frequency** of a simple pendulum:
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$
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\omega = \sqrt{\frac{g}{L}}
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$
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Where:
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- $\omega$ = angular frequency (rad/s)
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4. **Displacement of a simple pendulum (for small angles)**:
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$$
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\theta(t) = \theta_0 \cos(\omega t)
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$$
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Where:
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- $\theta(t)$ = angular displacement at time \(t\)
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- $\theta_0$ = maximum displacement (amplitude)
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5. **Energy of a simple pendulum**:
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- **Potential Energy** at maximum displacement:
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$$
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PE = mgh
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$$
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Where:
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- $m$ = mass of the pendulum bob
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- $h$ = height relative to the lowest point
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- **Kinetic Energy** at the equilibrium point:
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$$
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KE = \frac{1}{2}mv^2
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$$
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6. **Damped pendulum (under the influence of friction)**:
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$$
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\theta(t) = \theta_0 e^{-\gamma t} \cos(\omega t)
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$$
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Where:
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- $\gamma$ = damping coefficient
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7. **For a physical pendulum** (a rigid body swinging about a pivot):
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$$
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T = 2\pi \sqrt{\frac{I}{mgd}}
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$$
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Where:
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- $I$ = moment of inertia of the object
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- $d$ = distance from the pivot point to the center of mass
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These are the key formulas for general pendulum motion, covering simple and physical pendulums, damped motion, and energy analysis.
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# Dampened Springs
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The damping coefficient, denoted by $\gamma$, quantifies the rate at which the amplitude of oscillation decreases over time due to resistive forces (like friction or air resistance) acting on the system. It depends on factors like the properties of the medium (air, fluid, etc.) and the characteristics of the object (shape, surface area, etc.).
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The damping force is typically modeled as:
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$$
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F_{\text{damping}} = -\gamma v
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$$
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where:
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- $F_{\text{damping}}$ is the damping force,
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- $\gamma$ is the damping coefficient, and
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- $v$ is the velocity of the oscillating mass.
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The damping coefficient influences how quickly the system loses energy. The larger the damping coefficient, the faster the oscillations decay.
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### Types of Damping:
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1. **Underdamped**: If the damping is weak ($\gamma < \omega_0$), the system still oscillates, but with decreasing amplitude. The system's motion is described by:
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$$
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x(t) = A e^{-\gamma t} \cos(\omega' t + \phi)
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$$
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2. **Critically damped**: If the damping is strong enough to prevent oscillations, but not too strong to cause the system to return to equilibrium too slowly, we have the critical damping condition: $\gamma = \omega_0$.
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3. **Overdamped**: If the damping is too strong ($\gamma > \omega_0$), the system returns to equilibrium without oscillating, but more slowly than in the critically damped case.
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### Relation to Natural Frequency:
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The **damped angular frequency** $\omega'$ is related to the natural frequency $\omega_0$ by:
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$$
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\omega' = \sqrt{\omega_0^2 - \gamma^2}
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$$
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This shows that as $\gamma$ increases, $\omega'$ decreases, meaning the system oscillates less rapidly. If $\gamma$ is large enough (in the overdamped case), there will be no oscillations at all.
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In summary, the damping coefficient $\gamma$ plays a key role in determining the rate at which a mass-spring system loses energy and the nature of its oscillations.
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### Snell's Law:
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Snell's Law describes the relationship between the angles of incidence and refraction when a wave passes through the boundary between two media with different refractive indices. It is given by:
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$$
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n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
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$$
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Where:
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- $n_1$ and $n_2$ are the refractive indices of the two media,
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- $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively, measured from the normal.
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Snell's Law explains how light bends when it transitions from one medium to another (e.g., air to water). If the refractive index increases (e.g., from air to glass), the light slows down and bends towards the normal. Conversely, if the refractive index decreases (e.g., from water to air), light bends away from the normal.
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### Doppler Effect:
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The Doppler Effect refers to the change in frequency (or wavelength) of a wave as observed by someone moving relative to the wave source. For sound waves, the observed frequency $f'$ is given by:
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$$
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f' = f \left( \frac{v \pm v_{\text{observer}}}{v \pm v_{\text{source}}} \right)
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$$
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Where:
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- $f'$ is the observed frequency
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- $f$ is the emitted frequency,
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- $v$ is the wave speed in the medium,
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- $v_{\text{observer}}$ is the velocity of the observer relative to the medium,
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- $v_{\text{source}}$ is the velocity of the source relative to the medium.
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The sign conventions depend on whether the observer and source are moving towards or away from each other:
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- Moving towards each other: use the "+" sign in the numerator and the "-" in the denominator.
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- Moving apart: use the "-" sign in the numerator and the "+" in the denominator.
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In the case of light, the relativistic Doppler shift formula is used, where the frequency shift is influenced by both the motion and the speed of light.
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The Doppler Effect explains phenomena such as the change in pitch of a passing car or the redshift/blueshift observed in the light from stars and galaxies.
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