The curve \(C\) has parametric equations
\[x=\frac{1}{2} t^{2}-\ln t, \quad y=2 t+1, \quad \text { for } \frac{1}{2} \leqslant t \leqslant 2 .\]
(a) Find the exact length of \(C\).
(b) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\), simplifying your answer.