Problem $2)$

A pinned beam with a torsional $\backslash (\backslash$left$(\backslash$mathrm$\{$k$\}\_\{1\}\backslash$right$)\backslash )$ and linear $\backslash (\backslash$left$(\backslash$mathrm$\{$k$\}\_\{2\}\backslash$right$)\backslash )$ spring is shown in the figure. The beam is subjected to an end load of P$.$ Treat the EI for the material as a known value.

a$.$ What are the boundary conditions for the problem?

b$.$ Using the boundary conditions, find the equation to solve for the critical buckling load, you do not need solve it$.$

c$.$ Using the equation you develop in part $"$b$"$ and letting the torsional $(\backslash (\backslash$mathrm$\{$k$\}\_\{1\}\backslash ))$ spring go to zero as well as the linear spring going to infinity $\backslash (\backslash$left$(\backslash$mathrm$\{$k$\}\_\{2\}\backslash$right$)\backslash ),$ find the first critical buckling load explicitly. Comment on your results.