I have attached my work on the Focault Pendulum and I used Netwon's Notation for my derivatives. ### Foucault Pendulum Solution Using Complex SubstitutionIn this session, we have discussed that the solution of the Foucault pendulum can be obtained using a complex substitution \(\eta(t) = x(t) + iy(t)\). The solution is given by:\[ \eta(t) = e^{-i\Omega_z t} \left( C_1 e^{i \sqrt{\Omega_z^2 + \omega_0^2} t} + C_2 e^{-i \sqrt{\Omega_z^2 + \omega_0^2} t} \right)\]where \( C_1 \) and \( C_2 \) are complex constants.Using the initial conditions \( x_0 = A, y_0 = 0, v_{x_0} = 0 \), and \( v_{y_0} = 0 \), determine the real functions \( x(t) \) and \( y(t) \) and describe the pendulum's behavior by referencing the solutions (explain which part of the function is responsible for what). Recall that \( e^{i \theta} = \cos \theta + i \sin \theta \). ### Educational Content - Foucault Pendulum Analysis#### Diagrams Explanation:1. **Circular Path Diagram (Top Left Corner)**: - This represents the motion of a Foucault Pendulum as viewed from above the Earth. The pendulum swings back and forth while the Earth rotates beneath it, causing the plane of the pendulum's swing to appear to rotate. - **θ**: Angle of deviation. - **Fcf**: Centrifugal force due to Earth's rotation.2. **3D Coordinate System (Top Center)**: - Shows a 3-dimensional coordinate system with axes labeled \(x\), \(y\), and \(z\). The pendulum bob is shown in position with lengths \(L\) (length of the pendulum) and angles **θ** and **β** marked. - Forces acting on the pendulum bob are diagrammatically represented: tension \( T \) and gravitational force \( mg \).#### Mathematical Formulations:1. **Sum of Forces \( \Sigma F = m \ddot{\mathbf{r}} \)**: - The net force equation acting on the pendulum bob is derived with consideration of forces in different directions.2. **Force Components**: - **x-axis (East)**: \( \ddot{x} = -\frac{mgx}{L} + 2 \dot{y} \Omega_{S/LO} \cos(\theta) \) - **y-axis (North)**: \( \ddot{y} = -\frac{mgy}{L} + 2 \dot{x} \Omega_{S/LO} \cos(\theta) \) - Where **Ω** is the angular velocity of Earth's rotation.3. **Directional Cosines**: - Represented as a matrix involving the angular velocity vector \(\Omega_{S/LO}\).4. **Effective Gravitational Force \( g_{\text{eff}} \)**: - Simplifying to \( g_{\text{eff}} \approx g \).5. **Equations of Motion Simplified**: - \( \ddot{x}'' + \omega_0^2 x + 2 \Omega_2 \dot{y} = 0 \) - \( \ddot{y}'' + \omega_0^2 y + 2 \Omega_2 \dot{x}

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We have discussed in the session that the solution of the Foucault pendulum can be obtained using a complex substitution 7(t) = x(t)+ iy(t). The solution was 2. 2 n(t) = = e-in,t Ce + Cze where C, and Cz are complex constants. Using the initial conditions x, = A, yo = 0, vxo = 0, and vyo = 0, determine the real functions x (t) and y(t) and describe the pendulum's behavior be referencing the solutions(explain which part of the function is responsible for what). Recall that e16 = cos e + i sin 0.

We have discussed in the session that the solution of the Foucault pendulum can be obtained using a complex substitution 7(t) = x(t)+ iy(t). The solution was 2. 2 n(t) = = e-in,t Ce + Cze where C, and Cz are complex constants. Using the initial conditions x, = A, yo = 0, vxo = 0, and vyo = 0, determine the real functions x (t) and y(t) and describe the pendulum's behavior be referencing the solutions(explain which part of the function is responsible for what). Recall that e16 = cos e + i sin 0.