There are four rods A,B,C and D of same length l but different mass density d, 2d, 3d, 4d respectively. These are joined to form a square with sides C and D along x and y axis of coordinate axes respectively. Find the coordinate of centre of mass of structure.

 

Dear Student,

Please find below the solution to the asked query:

The coordinates of the centre of mass for each of the rod respectively are A (l/2, l); B (l, l/2); C(l/2, 0) & D(0, l/2).

The masses of the four rods are m, 2m, 3m & 4m​ respectively.

Then the common centre of masses are,

$$\begin{gathered} X_{cm}=\frac{m\left({\displaystyle \frac{l}{2}}\right)+2m\left(l\right)+3m\left({\displaystyle \frac{l}{2}}\right)+4m\left(0\right)}{m+2m+3m+4m}=\frac{4ml}{10m}=\frac{2l}{5} \\ {Y}_{cm}=\frac{m\left({\displaystyle l}\right)+2m\left({\displaystyle \frac{l}{2}}\right)+3m\left({\displaystyle 0}\right)+4m\left({\displaystyle \frac{l}{2}}\right)}{m+2m+3m+4m}=\frac{4ml}{10m}=\frac{2l}{5} \\ Therefore, the common centre of mass coordinates are, \\ \left(X_{cm},Y_{cm}\right)=\left(\frac{2l}{5},\frac{2l}{5}\right) \end{gathered}$$

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