[Kamog]

To weigh...

1 lb: put item on one side of beam and 1-lb weight on other side of balance
2 lb: put item + 1-lb weight on one side and 3-lb weight on other side
3 lb: put item on one side and 3-lb weight on other side
4 lb: put item on one side and 3-lb + 1-lb weights on other side
5 lb: put item + 1-lb + 3-lb weights on one side and 9-lb weight on other side
6 lb: put item + 3-lb weight on one side and 9-lb weight on other side
7 lb: put item + 3-lb weight on one side and 9-lb + 1-lb weight on other side
8 lb: put item + 1-lb weight on one side and 9-lb weight on other side
9 lb: put item on one side and 9-lb weight on other side
10 lb: put item on one side and 9-lb + 1-lb weights on other side
11 lb: put item + 1-lb weight on one side and 9-lb + 3-lb weights on other side
12 lb: put item on one side and 9-lb + 3-lb weights on other side
13 lb: put item one one side and 9-lb + 3-lb + 1-lb weights on other side
14 lb: put item + 9-lb + 3-lb + 1-lb weights on one side, 27-lb weight on other side
15 lb: put item + 9-lb + 3-lb weights on one side, 27-lb weight on other side
16 lb: put item + 9-lb + 3-lb weights on one side, 27-lb + 1-lb weights on other side
17 lb: put item + 9-lb + 1-lb weights on one side, 27-lb weight on other side
18 lb: put item + 9-lb weight on one side, 27-lb weight on other side
19 lb: put item + 9-lb weight on one side, 27-lb + 1-lb weights on other side
20 lb: put item + 9-lb + 1-lb weights on one side, 27-lb + 3-lb weights on other side
21 lb: put item + 9-lb weight on one side, 27-lb + 3-lb weights on other side
22 lb: put item + 9-lb weight on one side, 27-lb + 3-lb + 1-lb weights on other side
23 lb: put item + 3-lb + 1-lb weights on one side, 27-lb weights on other side
24 lb: put item + 3-lb weight on one side, 27-lb weight on other side
25 lb: put item + 3-lb weight on one side, 27-lb + 1-lb weights on other side
26 lb: put item + 1-lb weight on one side, 27-lb weight on other side
27 lb: put item on one side, 27-lb weight on other side
28 lb: put item on one side, 27-lb + 1-lb weight on other side
29 lb: put item + 1-lb weight on one side, 27-lb + 3-lb weights on other side
30 lb: put item on one side, 27-lb + 3-lb weights on other side
31 lb: put item on one side, 27-lb + 3-lb + 1-lb weights on other side
32 lb: put item + 3-lb + 1-lb weights on one side, 27-lb + 9-lb weights on other side
33 lb: put item + 3-lb weight on one side, 27-lb + 9-lb weights on other side
34 lb: put item + 3-lb weight on one side, 27-lb + 9-lb + 1-lb weights on other side
35 lb: put item + 1-lb weithg on one side, 27-lb + 9-lb weights on other side
36 lb: put item on one side, 27-lb + 9-lb weights on other side
37 lb: put item on one side, 27-lb + 9-lb + 1-lb weights on other side
38 lb: put item + 1-lb weight on one side, 27-lb + 9-lb + 3-lb weights on other side
39 lb: put item on one side, 27-lb + 9-lb + 3-lb weights on other side
40 lb: put item on one side, 27-lb + 9-lb + 3-lb + 1-lb weights on other side.

Therefore, you need to cut the 40-lb rock into 4 pieces: 27-lb, 9-lb, 3-lb, 1-lb.