Problem 4.6 Challenge

หาจุดศูนย์กลางมวลในแกน x

\(yx_{cm}=\frac{\int xdm}{M}\)

\(hx_{cm}=\frac{1}{M}\int xσdA\)

\(dm = σdA\)

\(x_{cm}=\frac{σ}{M}\int xydx\)

\(x_{cm}=\frac{M}{M\frac{1}{2}wh}\int x\frac{hx}{w}dx\)

\(_{w}\)

x

\(\frac{y =\frac{hx}{w}}{dxw}x_{cm}=\frac{2}{w^{2}}\int x^{2}dx\)

\(_{0}\)

x

\(x_{cm}=\frac{2w}{3}\)

หาจุดศูนย์กลางมวลในแกน y

\(yy_{cm}=\frac{\int ydm}{M}\)

\(dm = σdA\)

\(hy_{cm}=\frac{1}{M}\int yσdA\)

\(y_{cm}=\frac{σ}{M}\int y(w - x)dy\)

xw - x

\(y_{cm}=\frac{M}{M\frac{1}{2}wh}\int y(w -\frac{wy}{h})dy\)

\(x^{h}\)

\(\frac{y}{dxw}\)

\(y_{cm}=\frac{2}{h}\int (y-\frac{y^{2}}{h})dy\)

\(_{0}\)

\(y_{cm}=\frac{h}{3}\)

îĵ

\(\therefore \vec{r}_{cm}=\frac{2w}{3}+\frac{h}{3}\;\blacksquare\)