## Step 1: Write the Equation of Motion Let \( x \) be the displacement of the mass \( m \), and \( y = Y \cos(\omega t) \) be the position of the left end of the damper. - The force by the damper: \( F_{damping} = 3c(\dot{y} - \dot{x}) \) - The force by the spring: \( F_{spring} = -2k x \) The equation of motion using Newton's second law: \[ m\ddot{x} = F_{damping} + F_{spring} \] Substitute the expressions: \[ m\ddot{x} = 3c(\dot{y} - \dot{x}) - 2k x \] \[ m\ddot{x} + 3c\dot{x} + 2k x = 3c\dot{y} \] Since \( \dot{y} = -Y \omega \sin(\omega t) \): \[ m\ddot{x} + 3c\dot{x} + 2k x = -3c Y \omega \sin(\omega t) \] **Explanation:** We have derived the equation of motion by considering the forces due to the damper and spring and including the driving motion \( y \). --- ## Step 2: Find the Steady-State Amplitude Assume a steady-state solution of the form: \[ x_p = X \cos(\omega t - \phi) \] The right side is a sine, so let's rewrite as \( -3c Y \omega \sin(\omega t) = 3c Y \omega \cos(\omega t - \frac{\pi}{2}) \). For a driving force \( F_ \cos(\omega t - \delta) \), the amplitude is: \[ X = \frac{F_}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} \] Here, \( F_ = 3c Y \omega \) and \( \delta = \frac{\pi}{2} \). So, \[ X = \frac{3c Y \omega}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} \] **Explanation:** By matching the form of the forced oscillation equation, we identify the amplitude \( X \) of forced oscillation. --- ## Final Answer \[ \boxed{ X = \frac{3c Y \omega}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} } \] This is the amplitude of the forced oscillation for the displacement \( x \) of the object.

# Solution to Forced Oscillation Problem ## Step 1: Write the Equation of Motion Let \( x \) be the displacement of the mass \( m \), and \( y = Y \cos(\omega t) \) be the position of the left end of the damper. - **Damping Force:** The force exerted by the damper is given by: \[ F_{\text{damping}} = 3c(\dot{y} - \dot{x}) \] - **Spring Force:** The force exerted by the spring is: \[ F_{\text{spring}} = -2k x \] Using Newton's second law, the equation of motion can be expressed as: \[ m\ddot{x} = F_{\text{damping}} + F_{\text{spring}} \] Substituting the expressions for forces: \[ m\ddot{x} = 3c(\dot{y} - \dot{x}) - 2k x \] Rearranging gives: \[ m\ddot{x} + 3c\dot{x} + 2k x = 3c\dot{y} \] Now substituting \( \dot{y} = -Y \omega \sin(\omega t) \): \[ m\ddot{x} + 3c\dot{x} + 2k x = -3c Y \omega \sin(\omega t) \] **Explanation:** We have derived the equation of motion by considering the forces acting on the mass due to the damper and spring, with the displacement of the left end of the damper described as a harmonic function. --- ## Step 2: Find the Steady-State Amplitude Assuming a steady-state solution of the form: \[ x_p = X \cos(\omega t - \phi) \] The right side can be rewritten to match the cosine form: \[ -3c Y \omega \sin(\omega t) = 3c Y \omega \cos(\omega t - \frac{\pi}{2}) \] For a driving force \( F = 3c Y \omega \) acting as a cosine function, the amplitude \( X \) of the forced oscillation is given by: \[ X = \frac{F}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} \] Substituting \( F = 3c Y \omega \): \[ X = \frac{3c Y \omega}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} \] **Explanation:** By matching the form of the forced oscillation equation with the input driving force, we have derived the expression for the amplitude \( X \) of the oscillation. --- ## Final Answer \[ \boxed{ X = \frac{3c Y \omega}{\sqrt{(2k - m\omega^2)^2 + (3c\omega)^2}} } \] This represents the amplitude of the forced oscillation for the displacement \( x \) of the object.