有一厚度均勻的金屬板 其直徑為4米

有一厚度均勻的金屬板,直徑為D1=4米,若在圓板內挖去一個直徑為D2=2米的內切圓,則剩餘部份之質心距原來之質心幾米=?Sol:假設內切圓放置在x軸,圓心C=(1,0)y軸通過大圓圓心O=(0,0)大圓面積: A1=πD1^2/4=4π小圓面積: A2=πD2^2/4=π實際面積: A=A1-A2=3π各面積對x軸一次力矩:A*Xbar=A1*0-A2*1Xbar=-π/3π=-1/3(m)剩餘部份質心距原來質心=1/3(m)......ans


2013-01-06 05:18:53 補充：
各面積對x軸一次力矩:

修改為: 各面積對y軸一次力矩:

Let t be the thickness of the circular metal plate with density p.
Mass of original metal plate = (pi).(2^2).t.p = 4(pi)tp
Mass of cut-out circular plate = (pi).(1^2).t.p = (pi)tp
[where pi = 3.14159 ...]
Hence, mass of remaining plate = 4(pi)tp - (pi)tp = 3(pi)tp

Take moment about the centre of the original circular plate,
[3(pi)tp]x = (pi)tp.(1)
where x is the distance of the centre of mass of the remaining plate from the centre of mass of the original circular plate
solve for x gives x = 1/3 m