1. Draw and write the force equation of the linear translational system shown, and find the transfer function at figure a. and b. . Set Mg = 0 for the transfer functions. B2 M M ) + Mg R) + Mg (a) (b)

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In the pulley system shown in Figure P4.13, the input is the applied force f, and the output is the displacement x. Assume the pulley masses are negligible and derive the equation f motion. Figure P4.13 H Additionally, find the transfer function if f is the input and a is the output, and derive an expression for the natural frequency in terms of m, R, and k

2) Consider the spring-mass-damper system hanging down from a ceiling. Initially, the system is at rest. Then, we gently place another mass (M) on top the original one (m). First, please obtain the equation of motion of the mass m in the system (before we placed the other object). Assign the displacement references yourself and obtain the equation accordingly. From the equation of motion, please obtain the transfer function, G(s), between an external force on the mass m , f(t), and the displacement of the mass m, x(t). Then, please obtain the motion of the object M, using the transfer function G(s). Don't forget that G(s) belongs to the system shown on the top! To solve this, represent the force applied by the added mass (M) as a function of time. How would you describe the act of adding a mass as an external force input? Then solve for x(t). c = 10 Ns/m k = 50 N/m m = 5 kg M M = 25 kg F(s) G(s) X(s)

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7. A ball attached to a string of length 0.25 meters is whirled around in a circular path. It experiences a centripetal acceleration of 40 m/s². a. What is the tangential velocity of the ball? b. How long does the ball take to make one complete revolution?