A

Step 1: Understand the problem
We need to calculate the work necessary to overturn a crate of weight 1 ton (1000 kg) about two edges: AB and A'B'.

Step 2: Determine the dimensions of the crate
Let’s assume the crate has dimensions: height (h) = 80 cm, width (w) = 1 m (100 cm), and length (l) = 40 cm (given in the figure). The weight of the crate (W) = mg = 1000 kg * 9.81 m/s² = 9810 N.

Step 3: Calculate the height of the center of mass (CM)
For a rectangular crate, the center of mass is located at half the dimensions. Thus, the height of the CM from the base is 40 cm (h/2) = 40 cm.

Step 4: Calculate the moment arm for each edge
1. For edge AB (considering height about AB):
- Moment arm = distance from the center of mass to edge AB = 40 cm = 0.4 m.

2. For edge A'B':
- Moment arm = total height of the crate - distance to CM = 80 cm - 40 cm = 40 cm = 0.4 m.

Step 5: Calculate the work done to overturn the crate
The work done (W) is given by the formula:
$$W = F imes d$$
Where F is the weight (9810 N) and d is the height of the CM in each case (0.4 m).

For both edges:
$$ W_{AB} = 9810 imes 0.4 $$
$$ W_{AB} = 3924 ext{ J} $$
$$ W_{A'B'} = 9810 imes 0.4 $$
$$ W_{A'B'} = 3924 ext{ J} $$

Step 6: Conclusion
The work done for both edges is the same due to symmetry, thus the minimum amount of work requires to overturn the crate about either edge is equal.
Therefore, the correct answer is A.