b
Let the tension at point \(A\) be \(T_{A} .\) So, from Newton's second law 

\(T_{A}-m g=\frac{m v_{c}^{2}}{R}\)
Energy at point \(A=\frac{1}{2} m v_{0}^{2}\)   \(...(i)\)

Energy at point \(C\) is

\(\frac{1}{2} m v_{c}^{2}+m g \times 2 R \ldots .\)         \(...(ii)\)
Applying Newton's second law at point \(C\)

\(T_{c}+m g=\frac{m v_{c}^{2}}{R}\)
To complete the loop \(T_{c} \geq 0\)

So, \(m g=\frac{m v_{c}^{2}}{R}\)

\(\Rightarrow \quad v_{c}=\sqrt{g R}\)      \(...(ii)\)

From Eqs. \((i)\) and \((ii)\) by conservation of energy 

\(\frac{1}{2} m v_{0}^{2}=\frac{1}{2} m v_{c}^{2}+2 m g R\)

\(\Rightarrow \quad \frac{1}{2} m v_{0}^{2}=\frac{1}{2} m g R+2 m g R \quad\left(\because v_{c}=\sqrt{g R}\right)\)

\(\Rightarrow \quad v_{0}^{2}=g R+4 g R\)

\(\Rightarrow \quad v_{0}=\sqrt{5 g R}\)