C

To determine the condition required for a block to complete a full circular path, we can use conservation of mechanical energy and the centripetal force requirement.
Step 1: As the block descends from height h, it converts gravitational potential energy into kinetic energy. The potential energy at height h is given by: $$ PE = mgh $$ where m is the mass, g is the acceleration due to gravity, and h is the height.
Step 2: At the bottom of the circular path, this potential energy is converted into kinetic energy. The kinetic energy (KE) is given by: $$ KE = \frac{1}{2} mv^2 $$
By conservation of energy, we have: $$ mgh = \frac{1}{2} mv^2 $$
Canceling m gives us: $$ gh = \frac{1}{2} v^2 \implies v^2 = 2gh $$
Step 3: As the block moves in a circular path of radius r, it requires a centripetal force. The necessary centripetal acceleration is given by: $$ a_c = \frac{v^2}{r} $$
Substituting our expression for velocity gives: $$ a_c = \frac{2gh}{r} $$
Step 4: For the block to maintain its circular path, the gravitational force must provide the required centripetal force at the top of the circle. At the top of the path, the gravitational force is mg, and the required centripetal force is: $$ F_c = m \cdot a_c = m\frac{2gh}{r} $$
The condition for completing a full circle is that the gravitational force at least equals the required centripetal force: $$ mg \geq m\frac{2gh}{r} $$ \implies g \geq \frac{2gh}{r} \implies h \geq \frac{r}{2}
Conclusion: Hence, the relation that holds good to complete a full circle is:
$$ h > \frac{r}{2}
Therefore, option C is correct.