Let’s solve **part (a)**: *Rotational mechanical system 1.* --- ## **Step 1: Draw Free-Body Diagram and Write Torques for Each Inertia** ### For \( J_1 \): - **Torque due to spring \( k_1 \):** \( -k_1 \theta_1 \) - **Torque due to spring \( k_2 \):** \( -k_2 (\theta_1 - \theta_2) \) - **Torque due to damping \( b_1 \):** \( -b_1 \dot{\theta}_1 \) - **Net torque on \( J_1 \):** \[ J_1 \ddot{\theta}_1 = -k_1\theta_1 - k_2(\theta_1 - \theta_2) - b_1 \dot{\theta}_1 \] **Explanation:** Each term represents the restoring/damping torques from the springs and damper acting on \( J_1 \). The negative sign indicates that the force opposes displacement or velocity. --- ### For \( J_2 \): - **Torque due to spring \( k_2 \):** \( k_2(\theta_1 - \theta_2) \) - **Torque due to damping \( b_2 \):** \( -b_2 \dot{\theta}_2 \) - **Applied torque:** \( +T_a \) - **Net torque on \( J_2 \):** \[ J_2 \ddot{\theta}_2 = k_2(\theta_1 - \theta_2) - b_2 \dot{\theta}_2 + T_a \] **Explanation:** \( J_2 \) feels torque from the coupling spring, its own damper, and the external torque \( T_a \). --- ## **Step 2: Write the Equations of Motion** The coupled equations of motion are: \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + (b_1 + k_1 + k_2)\theta_1 - k_2\theta_2 + b_1\dot{\theta}_1 &= \\ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2\theta_2 - k_2\theta_1 &= T_a \\ \end{align*} } \] Or, more conventionally arranged: \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + b_1 \dot{\theta}_1 + (k_1 + k_2)\theta_1 - k_2\theta_2 &= \\ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2\theta_2 - k_2\theta_1 &= T_a \\ \end{align*} } \] **Explanation:** These are the equations of motion for \( \theta_1(t) \) and \( \theta_2(t) \), relating the angular displacements to the applied torque \( T_a \). Each inertia's equation accounts for all torques acting on it due to springs, dampers, and external forces. --- ### **Final Answer (for part a):** \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + b_1 \dot{\theta}_1 + (k_1 + k_2)\theta_1 - k_2\theta_2 &= \\ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2\theta_2 - k_2\theta_1 &= T_a \\ \end{align*} } \] --- *Let me know if you want part (b) solved similarly!*

Let's solve **part (b)**: *Rotational mechanical system 2.* --- ## **Step 1: Draw Free-Body Diagram and Write Torques for Each Inertia** ### For \( J_1 \) (Pendulum with spring): - **Torque due to spring \( k \):** \( -k \theta_1 \) - **Torque due to pendulum angle:** \( -mgL \sin(\theta_1) \approx -mgL \theta_1 \) (small angle approximation) - **Net torque on \( J_1 \):** \[ J_1 \ddot{\theta}_1 = -k \theta_1 - mgL \theta_1 \] **Explanation:** The torque from the spring and the weight of the pendulum both work against the angular displacement. The small angle approximation allows us to simplify the sine function. --- ### For \( J_2 \) (Coupled pendulum): - **Torque due to pendulum angle \( \theta_2 \):** \( -mgL \sin(\theta_2) \approx -mgL \theta_2 \) - **Coupling torque due to \( \theta_1 \):** \( -k(\theta_2 - \theta_1) \) - **Net torque on \( J_2 \):** \[ J_2 \ddot{\theta}_2 = -mgL \theta_2 - k(\theta_2 - \theta_1) \] **Explanation:** Similar to \( J_1 \), \( J_2 \) experiences restoring torque from its weight and the coupling spring, which connects it to \( J_1 \). --- ## **Step 2: Write the Equations of Motion** The coupled equations of motion are: \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + (k + mgL)\theta_1 - k\theta_2 &= 0 \\ J_2 \ddot{\theta}_2 + (mgL + k)\theta_2 - k\theta_1 &= 0 \\ \end{align*} } \] Or, more conventionally arranged: \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + (k + mgL)\theta_1 - k\theta_2 &= 0 \\ J_2 \ddot{\theta}_2 - k\theta_1 + (mgL + k)\theta_2 &= 0 \\ \end{align*} } \] **Explanation:** These equations of motion for \( \theta_1(t) \) and \( \theta_2(t) \) demonstrate the relationships between angular displacements of the pendulums under the influence of their respective springs and gravitational forces. --- ### **Final Answer (for part b):** \[ \boxed{ \begin{align*} J_1 \ddot{\theta}_1 + (k + mgL)\theta_1 - k\theta_2 &= 0 \\ J_2 \ddot{\theta}_2 - k\theta_1 + (mgL + k)\theta_2 &= 0 \\ \end{align*} } \] --- *If you need anything further, feel free to ask!*