Answer:
[tex]r_{cm}=[12.73,12.73]cm[/tex]
Explanation:
The general equation to calculate the center of mass is:
[tex]r_{cm}=1/M*\int\limits {r} \, dm[/tex]
Any differential of mass can be calculated as:
[tex]dm = \lambda*a*d\theta[/tex] Where "a" is the radius of the circle and λ is the linear density of the wire.
The linear density is given by:
[tex]\lambda=M/L=M/(a*\pi/2)=\frac{2M}{a\pi}[/tex]
So, the differential of mass is:
[tex]dm = \frac{2M}{a\pi}*a*d\theta[/tex]
[tex]dm = \frac{2M}{\pi}*d\theta[/tex]
Now we proceed to calculate X and Y coordinates of the center of mass separately:
[tex]X_{cm}=1/M*\int\limits^{\pi/2}_0 {a*cos\theta*2M/\pi} \, d\theta[/tex]
[tex]Y_{cm}=1/M*\int\limits^{\pi/2}_0 {a*sin\theta*2M/\pi} \, d\theta[/tex]
Solving both integrals, we get:
[tex]X_{cm}=2*a/\pi=12.73cm[/tex]
[tex]Y_{cm}=2*a/\pi=12.73cm[/tex]
Therefore, the position of the center of mass is:
[tex]r_{cm}=[12.73,12.73]cm[/tex]
