# How to find the center of mass of thin flat plate?

I want to find the center of mass of thin flat plate with constant density $$\delta=3g/cm^{-2}$$ as shown in figure. I know that center of mass formula is: $$(x,y)=\bigg(\frac{\int xdm}{\int dm},\frac{\int ydm}{\int dm}\bigg)$$ For $$x-$$coordinate of the center of mass, firstly I consider a vertical strip and then compute the moment of the vertical strip about $$y-axis$$ as:

Length of the vertical strip$$=2x$$,

width of the vertical strip$$=dx$$,

area of the vertical strip$$=dA=2xdx$$, and

mass of the vertical strip$$=dm=6xdx$$

The moment of the strip about $$y-axis$$ is: $$xdm=6x^2dx$$

The moment of the plate about $$y-axis$$ is therefore: $$M_{y}=\int_{0}^{1}6x^2 dx=2g.cm$$

The mass of the plate is: $$M=\int_{0}^{1}6x^2dx=3g$$, so the $$x-$$coordinate of the center of mass is: $$x^{'}=\frac{2}{3}cm.$$

My question is how can I compute the $$y-$$coordinate of the center of mass for this thin flat plate using this vertical strip?

I tried something as:

The $$y-$$component of the vertical strip center of mass$$=\frac{y}{2}=x$$

The moment of the plate about $$x-$$axis$$M_{x}=\int_{0}^{1}\frac{y}{2}dm=\int_{0}^{1}6x^2dx=2g.cm$$ Then $$y-$$coordinate of the center of mass$$=\frac{2}{3}cm$$

Am I right?

• It's a triangle: you could use the centroid – b00n heT May 27 '19 at 19:07
• @b00nheT Sir I have no information about centroid, How can I use the integral formula for finding the y coordinate of the center of mass written in question. – Noor Aslam May 27 '19 at 19:15

The center of mass of a triangle is the intersection of the medians. Its coordinates are the averages of the coordinates of the vertices. In this case the centroid is at $$(2/3, 2/3)$$, just as in your picture.
• Sir you are cent percent write but how can I get the $y-$coordinate of the center of mass through integral approach? – Noor Aslam May 27 '19 at 19:26