Problem 1-3
See solutions 2-4.
Problem 4
Following the discussion in class, we first get the bending moment function
\[M(x) = -Fl+Fx\]Consider changing cross-sections
\[M/I = -\frac{Fl}{I_1} + \frac{Fx}{I_1} + (\frac{F}{I_2}-\frac{F}{I_1})<x-l/2>^1 + (-\frac{Fl}{2I_2}+\frac{Fl}{2I_1})<x-l/2>^0\]With two integrations and using the conditions \(\theta(0) = 0\) and \(y(0)=0\), we get
\(y(l) = -\frac{Fl^3}{24}(1/I_2 + 7/I_1)\) or \(y(l) \propto 1/I_2 + 7/I_1\)
Notice that \(I \propto r^4\) where \(r\) is the radius of the circular cross-section, and that \(r_1^2 + r_2^2 = C\) where \(C\) is a constant, we get
\(y(l) \propto (C-a)^{-2} + 7a^{-2}\), where \(a = r_1^2\). Thus \(y(l)\) reaches the minimal deflection when \(a = \frac{C (7^{1/3})}{1+(7^{1/3})}\).