4. (35%) The buckling equation for the column shown in Figure 3, where $\beta_R$ denotes the stiffness of the rotational spring at end B, is given as:
$\frac{\beta_R L}{EI}(kL \cot kL - 1) - k^2 L^2 = 0$.
(a) Derive the buckling equation. [Hint: introduce $\theta_B$ and find the support reactions]
(b) Determine the critical load for the case where $\beta_R \to \infty$.