Let's go through each part of the problem step by step. --- ## **Given Data and Setup** - **Heliostat mass:** \( m \) - **Heliostat length:** \( L \) - **Moment of inertia about the pivot:** \( J \) - **Distance from pivot to center of mass:** \( \ell \) - **Angle:** \( \theta(t) \), measured from horizontal axis - **Torque applied at pivot:** \( T(t) \) - **Gravity acts downwards:** \( g \) --- ## **(a) Differential Equation for the Nonlinear Dynamics** ### **Step 1: Free Body Diagram and Torques** - **Torque from gravity** (about the pivot): The force due to gravity acts at the center of mass, a distance \( \ell \) from the pivot. - Magnitude: \( mg \) - Perpendicular distance (moment arm): \( \ell \sin\theta \) - **Torque due to gravity:** \( -mg \ell \sin\theta \) (negative sign because gravity tries to decrease \( \theta \) if \( \theta > \)) - **Applied torque:** \( T(t) \) (positive in the direction of increasing \( \theta \)) ### **Step 2: Equation of Motion** Using Newton's second law for rotation: \[ J \ddot{\theta}(t) = \text{Sum of torques about the pivot} \] \[ J \ddot{\theta}(t) = T(t) - mg\ell \sin\theta(t) \] Or, rearranged: \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] --- ## **(b) Find \( T_ \) to Hold at Constant Angle \( \theta_ \)** If we want \( \theta(t) = \theta_ \) (constant), then \( \dot{\theta} = \), \( \ddot{\theta} = \). Plug into the equation: \[ J \cdot + mg\ell \sin\theta_ = T_ \] \[ T_ = mg\ell \sin\theta_ \] --- ## **(c) Linearize Around \( \theta_ \) and Find the Linearized Equation** Let \( \theta(t) = \theta_ + \delta\theta(t) \), with \( \delta\theta(t) \) small. Expand \( \sin\theta \) using Taylor expansion: \[ \sin\theta \approx \sin\theta_ + \cos\theta_ \cdot \delta\theta \] Substitute into the original equation: \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] \[ J \delta\ddot{\theta}(t) + mg\ell (\sin\theta_ + \cos\theta_ \cdot \delta\theta) = T_ + \delta T(t) \] Recall \( T_ = mg\ell \sin\theta_ \): \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] Or, \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] This is a linear second-order ODE for small deviations around \( \theta_ \). --- ## **(d) Nature of the Dynamics: Is It Underdamped or Overdamped?** ### **Inspection** - The linearized equation (without any damping term) is: \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] - The system is **undamped** because there is **no damping term** (no term proportional to \( \delta\dot{\theta} \)). - The solution will be **oscillatory** (if \( \cos\theta_ > \)), with natural frequency: \[ \omega_n = \sqrt{\frac{mg\ell \cos\theta_}{J}} \] - **Gravity alone does not provide damping**; it only acts as a restoring (or destabilizing) force depending on \( \theta_ \). - **Damping** (mechanical friction, air resistance, etc.) would appear as a term like \( +b \delta\dot{\theta}(t) \) in the differential equation, which is absent here. --- ## **Summary of Final Answers** ### **(a) Nonlinear Equation of Motion:** \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] ### **(b) Torque Needed to Hold at \( \theta_ \):** \[ T_ = mg\ell \sin\theta_ \] ### **(c) Linearized Equation Around \( \theta_ \):** \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] ### **(d) Nature of Motion (Damping):** - The equation is **undamped** (no damping term). - Gravity provides only a restoring force (if \( \cos\theta_ > \)), not damping. - Any damping must come from physical sources like friction or air resistance, **not gravity**. --- Let me know if you need any further explanation or derivations!

# Motion of a Heliostat Subject to Gravity This document outlines the step-by-step solution to the problem of a heliostat's motion, focusing on deriving the differential equations governing its dynamics. ## **Given Data and Setup** - **Mass of heliostat:** \( m \) - **Length of heliostat:** \( L \) - **Moment of inertia about the pivot:** \( J \) - **Distance from pivot to center of mass:** \( \ell \) - **Angle:** \( \theta(t) \) (measured from the horizontal axis) - **Torque applied at pivot:** \( T(t) \) - **Acceleration due to gravity:** \( g \) --- ## **(a) Differential Equation for the Nonlinear Dynamics** ### **Step 1: Free Body Diagram and Torques** 1. **Torque due to gravity:** - The force due to gravity acts at the center of mass, a distance \( \ell \) from the pivot. - Magnitude: \( mg \) - Perpendicular distance (moment arm): \( \ell \sin\theta \) - **Torque from gravity:** \[ \tau_g = -mg\ell \sin\theta \] (Negative sign denotes that gravity tends to decrease \( \theta \).) 2. **Applied torque:** - \( T(t) \) (positive in the direction of increasing \( \theta \)). ### **Step 2: Equation of Motion** Using Newton's second law for rotation: \[ J \ddot{\theta}(t) = T(t) - mg\ell \sin\theta(t) \] Rearranging gives us the nonlinear equation of motion: \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] --- ## **(b) Find \( T_ \) to Hold at Constant Angle \( \theta_ \)** To maintain a constant angle \( \theta(t) = \theta_ \), we set: - \( \dot{\theta} = 0 \) - \( \ddot{\theta} = 0 \) Substituting into the equation: \[ J \cdot 0 + mg\ell \sin\theta_ = T_ \] Thus, the required torque \( T_ \) to maintain the position is: \[ T_ = mg\ell \sin\theta_ \] --- ## **(c) Linearize Around \( \theta_ \) and Find the Linearized Equation** Let \( \theta(t) = \theta_ + \delta\theta(t) \), where \( \delta\theta(t) \) is small. ### **Taylor Expansion of \( \sin\theta \)** Using the Taylor series expansion around \( \theta_ \): \[ \sin\theta \approx \sin\theta_ + \cos\theta_ \cdot \delta\theta \] ### **Substituting into the Original Equation** The original equation becomes: \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] Substituting the Taylor expansion: \[ J \delta\ddot{\theta}(t) + mg\ell \left( \sin\theta_ + \cos\theta_ \cdot \delta\theta \right) = T_ + \delta T(t) \] Since \( T_ = mg\ell \sin\theta_ \): \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] This leads us to the linearized equation: \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] --- ## **(d) Nature of the Dynamics: Is It Underdamped or Overdamped?** ### **Analysis of the Linearized Equation** The linearized equation is: \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] - **Damping:** The equation lacks a damping term, which would typically be of the form \( b \delta\dot{\theta}(t) \). Therefore, the system is **undamped**. - **Oscillatory Behavior:** The system will exhibit oscillatory motion if \( \cos\theta_ > 0 \). The natural frequency of oscillation is given by: \[ \omega_n = \sqrt{\frac{mg\ell \cos\theta_}{J}} \] - **Gravity's Role:** Gravity acts as a restoring force that attempts to bring the heliostat back to its equilibrium position. It does not provide damping. --- ## **Summary of Final Answers** ### **(a) Nonlinear Equation of Motion:** \[ J \ddot{\theta}(t) + mg\ell \sin\theta(t) = T(t) \] ### **(b) Torque Needed to Hold at \( \theta_ \):** \[ T_ = mg\ell \sin\theta_ \] ### **(c) Linearized Equation Around \( \theta_ \):** \[ J \delta\ddot{\theta}(t) + mg\ell \cos\theta_ \cdot \delta\theta = \delta T(t) \] ### **(d) Nature of Motion (Damping):** - The equation is **undamped** (no damping term). - Gravity provides only a restoring force, not damping. - Any damping must come from external factors (e.g., friction). --- If further clarification or additional calculations are needed, feel free to ask!