A

(i) To find the minimum height h required for the body to complete the loop:
In order for the body to successfully complete the loop of radius r, it needs enough gravitational potential energy to convert into kinetic energy, ensuring that at the top of the loop, the centripetal force is equal to the gravitational force acting on it.

• At the top of the loop, the centripetal force required is given by:
  $$ F_c = \frac{mv^2}{r} $$
• The gravitational force acting on the block at the top of the loop is:
  $$ F_g = mg $$
• For the block to stay in motion, we require:
  $$ F_c \geq F_g $$
  or,
  $$ \frac{mv^2}{r} \geq mg $$
  which simplifies to:
  $$ v^2 \geq rg $$

Using conservation of energy:
The total mechanical energy at height h is equal to the total mechanical energy at the top of the loop (height 2r):
$$ mgh = mg(2r) + \frac{1}{2}mv^2 $$
Substituting for v from the previous inequality,
$$ mgh = mg(2r) + \frac{1}{2}m(\frac{rg}{m}) $$
After simplifying, we find:
$$ h = 5r $$
This is the minimum height required to complete the loop.
(ii) If h is double that minimum height:
If h = 10r, we check the total mechanical energy when it reaches the top of the loop:
Using energy conservation at height h:
$$ mgh = mg(2r) + \frac{1}{2}mv^2 $$ with h = 10r:
$$ mg(10r) = mg(2r) + \frac{1}{2}mv^2 $$
which gives us:
$$ v^2 = 16gr $$
At the top of the loop, the resultant force (R) on the block is given by:
$$ R = F_g - F_c $$
Substituting for F_g and F_c:
$$ R = mg - \frac{mv^2}{r} \to R = mg - \frac{m(16gr)}{r} $$
which simplifies to:
$$ R = mg - 16mg = -15mg $$
Thus, the block is exerting a downward force of fifteen times its weight (suggesting that it would not just maintain contact, but would be subjected to an increasing force). The negative sign indicates that the gravitational force is significantly less than the centripetal force required, thus ensuring the body stays on the track excitedly. However, since we are looking for resultant forces and magnitudes, the absolute value would suffice. Therefore, the final resultant force on the block at position H is:
$$ |R| = 15mg $$.