1. Consider the inverted pendulum illustrated in the following figure.
   1. (5 points) Determine the equation of motion for the system.
   2. (5 points) Determine the transfer function from the input torque to the output angle, i.e., find $\frac{\Theta(s)}{\tau(s)}$.
   3. (5 points) Determine the transfer function from the input torque to the output angular velocity, i.e., find $\frac{s \Theta(s)}{\tau(s)}$.
   4. Assume the torque is supplied by a d.c. motor, as illustrated in the following figure.
      1. (5 points) Determine the transfer function from the applied voltage to the output angle, i.e., find $\frac{\Theta(s)}{E(s)}$.
      2. (5 points) Determine the transfer function from the applied voltage to the output angular velocity, i.e., find $\frac{s \Theta(s)}{E(s)}$.

2. Consider the mass-pulley system illustrated in the following figure.
   1. (5 points) Determine the equation of motion for the system.
   2. (5 points) Determine the transfer function from the torque to the position of the mass, i.e., determine $\frac{\Theta(s)}{\tau(s)}$.
   3. (5 points) Determine the transfer function from the torque to the velocity of the mass, i.e., determine $\frac{s \Theta(s)}{\tau(s)}$.
   4. Assume the torque is supplied by a d.c. motor, as illustrated in the following figure.
      1. (5 points) Determine the transfer function from the applied voltage to the position of the mass, i.e., find $\frac{X(s)}{E(s)}$.
      2. (5 points) Determine the transfer function from the applied voltage to the velocity of the mass, i.e., find $\frac{s X(s)}{E(s)}$.
      3. (5 points) Determine the transfer function from the current in the circuit to the position of the mass, i.e., find $\frac{X(s)}{I(s)}$.
      4. (5 points) Determine the transfer function from the current in the circuit to the velocity of the mass, i.e., find $\frac{s X(s)}{I(s)}$.
3. (10 points) Find the transfer function $\frac{Y(s)}{R(s)}$ for the following block diagram.
4. (10 points) Find the transfer function $\frac{Y(s)}{R(s)}$ for the following block diagram.