如何計算重心的位置?

The most common method is to take moment about the new centre of mass
In your example, let
L be the length of string
mg be the weight of the string
Mg be the weight of the suspended object
The centre of mass of the string is at the middle of the length of the string, i.e. L/2 from each end.
In that case, it is expected that the new centre of mass of the string and the object combined together is somewhere between the centre of mass of the string at L/2 and the lower end of the string.
Let the position of the new centre of mass be x from the lower end of the string. Hence, taking moment about this new centre of mass,
Mg.(x) = mg.(L/2-x)
thus, Mx = mL/2 - mx
i.e. x = mL/2(M+m)
Therefore, the new centre of mass is at a distance of mL/2(M+m) from the lower end of the string.

質量中心或稱質心，指物質系統上被認為質量集中於此的一個假想點。與重心不同的是，質心不一定要在有重力場的系統中。值得注意的是，除非重力場是均勻的，否則同一物質系統的質心與重心不通常在同一假想點上。對於密度均勻、形狀對稱分佈的物體，其質心位於其幾何中心處。
在一個N維空間中的質量中心，座標系計算公式為：

圖片參考：http://upload.wikimedia.org/math/7/d/1/7d1511385aef4cfa4a2c8863d0c4b66f.png
X表示某一座標軸mi 表示物質系統中，某i質點的質量xi 表示物質系統中，某i質點的座標。
mi 表示物質系統中，某i質點的質量