Let's solve **each model** (Figures 2.35–2.38) step-by-step: --- ## **Figure 2.35: Vertical Oscillation with Two Springs** ### **(a) Newton’s Second Law** **Free Body Diagram:** - Upward force by upper spring: \( kx \) - Downward force by lower spring: \( kx \) - Weight: \( mg \) **Equation of Motion:** At equilibrium, \( mg = 2k x_0 \) (where \( x_0 \) is static deflection). Let \( x(t) \) be displacement from equilibrium. By Newton’s law: \[ m \frac{d^2x}{dt^2} = -k x(t) - k x(t) = -2k x(t) \] \[ \boxed{m \ddot{x} + 2k x = 0} \] **Linear/Nonlinear:** Linear **Natural Frequency:** \[ \omega_n = \sqrt{\frac{2k}{m}} \] --- ### **(b) Energy Method** Total energy \( E = \) Kinetic + Potential: \[ E = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}(2k)x^2 \] Set \( \frac{dE}{dt} = 0 \Rightarrow m \ddot{x} + 2k x = 0 \) --- ## **Figure 2.36: Horizontal Oscillation on Frictionless Surface** ### **(a) Newton’s Second Law** **Free Body Diagram:** - Rightward force by spring: \( -k x \) Equation: \[ m \frac{d^2x}{dt^2} = -k x \] \[ \boxed{m \ddot{x} + k x = 0} \] **Linear/Nonlinear:** Linear (for small \( x \)), but nonlinear if impact with wall. **Natural Frequency:** \[ \omega_n = \sqrt{\frac{k}{m}} \] --- ### **(b) Energy Method** \[ E = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}k x^2 \] Again, \( \frac{dE}{dt} = 0 \Rightarrow m \ddot{x} + k x = 0 \) --- ## **Figure 2.37: Simple Pendulum** ### **(a) Newton’s Second Law** **Free Body Diagram:** - Tangential force: \( -mg \sin \theta \) \[ m l \frac{d^2 \theta}{dt^2} = -mg \sin \theta \] \[ \boxed{l \ddot{\theta} + g \sin \theta = 0} \] **Linear/Nonlinear:** Nonlinear. For small \( \theta \), \( \sin\theta \approx \theta \): \[ l \ddot{\theta} + g \theta = 0 \] **Natural Frequency:** \[ \omega_n = \sqrt{\frac{g}{l}} \] --- ### **(b) Energy Method** \[ E = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l (1 - \cos \theta) \] For small \( \theta \), \( 1-\cos\theta \approx \frac{1}{2}\theta^2 \) --- ## **Figure 2.38: Torsional Oscillator** ### **(a) Newton’s Second Law** **Free Body Diagram:** - Restoring torque: \( -K \theta \) - Moment of inertia: \( I \) \[ I \frac{d^2\theta}{dt^2} = -K \theta \] \[ \boxed{I \ddot{\theta} + K \theta = 0} \] **Linear/Nonlinear:** Linear **Natural Frequency:** \[ \omega_n = \sqrt{\frac{K}{I}} \] --- ### **(b) Energy Method** \[ E = \frac{1}{2} I \dot{\theta}^2 + \frac{1}{2} K \theta^2 \] --- ## **Summary Table** | Figure | Equation of Motion | Linear? | Natural Frequency | |--------|-----------------------------------|------------|----------------------------| | 2.35 | \( m\ddot{x} + 2k x = 0 \) | Linear | \( \sqrt{\frac{2k}{m}} \) | | 2.36 | \( m\ddot{x} + k x = 0 \) | Linear* | \( \sqrt{\frac{k}{m}} \) | | 2.37 | \( l\ddot{\theta} + g \sin\theta = 0 \) | Nonlinear | \( \sqrt{\frac{g}{l}} \) (small \(\theta\)) | | 2.38 | \( I\ddot{\theta} + K \theta = 0 \) | Linear | \( \sqrt{\frac{K}{I}} \) | (*linear until impact*) --- **Let me know if you need free-body diagrams or further clarification on any part!**

Let's analyze each model step-by-step, drawing free-body diagrams, deriving equations of motion using Newton's second law, and applying the energy method. We'll also determine if the oscillations are linear or nonlinear and find the natural frequencies. --- ### **Figure 2.35: Vertical Oscillation of a Mass \( m \) with Two Springs** #### **(a) Free-Body Diagram:** - The mass \( m \) is attached to two springs, one on top and one below, both with spring constant \( k \). - When displaced downward by \( x \), the springs stretch, exerting restoring forces upward: \( kx \) from each spring. - The weight acts downward: \( mg \). **Diagram:** ``` Spring (top) Mass Spring (bottom) | m | | | | --- | --- (k) | (k) ``` --- #### **(a) Equation of Motion via Newton's Second Law:** - **Assumption:** \( x(t) \) is the displacement from equilibrium. - **At equilibrium:** The springs balance the weight: \[ 2k x_0 = mg \Rightarrow x_0 = \frac{mg}{2k} \] - **Displacement from equilibrium:** \( x(t) \). Total restoring force: \[ F_{restoring} = -2k x(t) \] - **Newton's law:** \[ m \frac{d^2x}{dt^2} = -2k x \] **Equation:** \[ \boxed{ m \ddot{x} + 2k x = 0 } \] --- #### **(b) Energy Method:** - **Total energy:** Kinetic + Potential \[ E = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} (2k) x^2 \] - **Equation of motion** from energy conservation: \[ \frac{dE}{dt} = 0 \Rightarrow m \ddot{x} + 2k x = 0 \] --- #### **Linear or Nonlinear?** - The restoring force is proportional to \( x \), so **linear oscillations**. #### **Natural Frequency:** \[ \boxed{ \omega_n = \sqrt{\frac{2k}{m}} } \] --- ### **Figure 2.36: Horizontal Sliding Oscillation of Mass \( m \) on a Frictionless Surface** #### **(a) Free-Body Diagram:** - The mass \( m \) slides on a frictionless surface. - When displaced by \( x \), the spring exerts a restoring force \( -k x \). **Diagram:** ``` Mass (m) --- Spring --- Wall ``` --- #### **(a) Equation of Motion via Newton's Second Law:** \[ m \ddot{x} = -k x \] **Equation:** \[ \boxed{ m \ddot{x} + k x = 0 } \] #### **(b) Energy Method:** \[ E = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2 \] --- #### **Linear or Nonlinear?** - For small \( x \), **linear oscillations**. - Impact with the wall (if relevant) introduces **nonlinearity** when oscillations are large. #### **Natural Frequency:** \[ \boxed{ \omega_n = \sqrt{\frac{k}{m}} } \] --- ### **Figure 2.37: Simple Pendulum of Length \( l \) and Mass \( m \)** #### **(a) Free-Body Diagram:** - Tangential component of gravity provides restoring torque: \[ \text{Restoring torque} = -mg l \sin \theta \] - Moment of inertia for small oscillations: \[ I = m l^2 \] **Equation of motion:** \[ I \ddot{\theta} = -mg l \sin \theta \] \[ m l^2 \ddot{\theta} + mg l \sin \theta = 0 \] Simplify: \[ l \ddot{\theta} + g \sin \theta = 0 \] --- #### **(a) Equation of Motion:** \[ \boxed{ l \ddot{\theta} + g \sin \theta = 0 } \] **Linear or Nonlinear?** - **Nonlinear** in general. - **Linearized** for small angles (\( \sin \theta \approx \theta \)): \[ l \ddot{\theta} + g \theta = 0 \] **Natural frequency (small oscillations):** \[ \boxed{ \omega_n = \sqrt{\frac{g}{l}} } \] --- #### **(b) Energy Method:** \[ E = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l (1 - \cos \theta) \] --- ### **Figure 2.38: System Under Torsional Vibration** #### **(a) Free-Body Diagram:** - Torsional restoring torque: \( -K \theta \) - Moment of inertia: \( I \) **Equation:** \[ I \ddot{\theta} + K \theta = 0 \] --- #### **(a) Equation of Motion:** \[ \boxed{ I \ddot{\theta} + K \theta = 0 } \] --- #### **(b) Energy Method:** \[ E = \frac{1}{2} I \dot{\theta}^2 + \frac{1}{2} K \theta^2 \] --- ### **Summary of Results:** | Model | Equation of Motion | Oscillation Type | Natural Frequency \( \omega_n \) | |---------|----------------------|-------------------|------------------------------| | **Figure 2.35** | \( m \ddot{x} + 2k x = 0 \) | Linear | \( \sqrt{\frac{2k}{m}} \) | | **Figure 2.36** | \( m \ddot{x} + k x = 0 \) | Linear (small oscillations) | \( \sqrt{\frac{k}{m}} \) | | **Figure 2.37** | \( l \ddot{\theta} + g \sin \theta = 0 \) | Nonlinear | \( \sqrt{\frac{g}{l}} \) (small \( \theta \)) | | **Figure 2.38** | \( I \ddot{\theta} + K \theta = 0 \) | Linear | \( \sqrt{\frac{K}{I}} \) | --- **Note:** For large amplitude oscillations in the pendulum, the motion is nonlinear, and the simple harmonic approximation no longer holds exactly, but the above linearized frequency gives an approximate natural frequency. --- If you'd like, I can help you draw the free-body diagrams or clarify any specific step further!