(Solved): Solve the following simple mechanical system. The applied force \( f(t) \) on a rigid bar, attache ...

Solve the following simple mechanical system. The applied force \( f(t) \) on a rigid bar, attached to a mass \( m \), can push or pull in both directions, i.e., \( \pm x \), and has no rotational component. Consider this system, with \( m=20 \mathrm{~kg} \) and \( k=5 \mathrm{~N} / \mathrm{m} \). i) Find the position of the mass, \( x(t) \), for the following conditions: \( \boldsymbol{f}(\boldsymbol{t})=0 \mathrm{~N} ; \boldsymbol{x}(0)=4 \mathrm{~m} ; \boldsymbol{v}(0)=4 \mathrm{~m} / \mathrm{s} \) Does the mass ever reach a steady state position? Explain. ii) Find the velocity of the mass, \( v(t) \), for the following conditions: \( \boldsymbol{f}(\boldsymbol{t})=150 \mathrm{~N} \cos (2 t) ; \quad \boldsymbol{x}(0)=10 \mathrm{~m} ; \quad \frac{d x}{d t}(0)=0 \mathrm{~m} / \mathrm{s} \) This is a linear system, which means that the output should have the same form as the input. For a simple, sinusoidally varying input force, the system should eventually respond with a simple, sinusoidally varying output velocity at the same frequency. Why doesn't that happen?