A uniform ladder 10 m long, weighing 500N, lends against a vertical wall making an angle of 30 degrees from the horizontal wall. A man weighing 800N climbs up the ladder to a point vertical distance of 3 m above the floor. Where is the center of gravity of the system consisting the man and the ladder.

To find the center of gravity for two objects, given that you know the positions of the center of gravity of each, you do the following: add the sum of each object's weight times its distance from some chosen point, then divide that sum by the total weight. This gives you the distance from your chosen point to the CG:

CG_total = (CG_1 x W_1 + CG_2 x W_2)/(W_1+W_2)

In this problem you can assume that the CG for the ladder is half way along it's length, or at 5 meters as measured from the point where the ladder is touching the floor. The man is positioned 3m off the ground, and you need to determine how far along the ladder he is, which you can determine by applying the appropriate trigonometry function. You then will have the two distances as measured along the ladder from where it touches the floor, and the two weights - can you take it from here?