### Understanding Harmonic Motion with DampingA mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57 N when the mass has a velocity of \( 6 \frac{m}{s} \). The viscous resistance is proportional to the speed of the object.Suppose the object is displaced an additional 0.07 m and released.Find a function to express the object's displacement from the spring's natural position, in meters after \( t \) seconds. Let positive displacements indicate a stretched spring, and use \( 9.8 \frac{m}{s^2} \) as the acceleration due to gravity.\[ u(t) = \boxed{} \]The given problem involves the following important details:1. **Mass:** 0.75 kg.2. **Initial Stretch of Spring:** 0.05 m.3. **Viscous Resistance:** 57 N at a velocity of \( 6 \frac{m}{s} \).4. **Additional Displacement:** 0.07 m.5. **Gravitational Acceleration:** \( 9.8 \frac{m}{s^2} \).To find the function \( u(t) \), we need to consider the system's damping and harmonic motion properties, which typically lead to a differential equation describing the mass-spring system.### Steps to Solve:1. **Determine Spring Constant (k):** - \( mg = kx \Rightarrow k = \frac{mg}{x} = \frac{0.75 \times 9.8}{0.05} = 147 \, N/m \).2. **Determine Damping Coefficient (c):** - \( c = \frac{57 N}{6 \frac{m}{s}} = 9.5 \frac{Ns}{m} \).3. **Formulate the Differential Equation:** - The equation of motion for a damped harmonic oscillator: \[ m\ddot{u} + c\dot{u} + ku = 0 \]. - Plug in the values: \[ 0.75\ddot{u} + 9.5\dot{u} + 147u = 0 \].4. **Initial Conditions:** - \( u(

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A mass of 0.75 kg stretches a spring 0.05 m. The mass is in a medium that exerts a viscous resistance of 57 m N when the mass has a velocity of 6 -. The viscous resistance is proportional to the speed of the object. Suppose the object is displaced an additional 0.07 m and released. m Find an function to express the object's displacement from the spring's natural position, in m after t seconds. Let positive displacements indicate a stretched spring, and use 9.8 as the acceleration due to 8² gravity. u(t) =