91Ó°ÊÓ

$6d=6m$

Use a basic formula for the center of mass. Centre of mass is the point, which represent the mean position of a matter of a body.

Formula is as follow:

$x_{cm}=\frac{x_1{m}_1+x_2{m}_2}{m_1+m_2}$

where, m₁, m₂ are masses, x₁, x₂ are points where m₁, m₂ are present and x_cm is centre of mass.

Suppose, negative mass is added of -m kg at the appropriate location. Then, total mass M kgis found as follows:

$Mass=density×volume$

$M=pAL$

Since, density and thickness is uniform, mass depends only on area.

So, mass of the added plate is as follows:

$\mathrm{m}={\left(2\mathrm{m}\right)}^2$

And, mass of whole plate is,

$$\begin{gathered} \mathrm{M}={\left(6\mathrm{m}\right)}^2 \\ \frac{\mathrm{m}}{\mathrm{M}}=\frac{4}{36} \\ \mathrm{M}=9\mathrm{m} \end{gathered}$$

Now, x component of center of mass is as follows:

$$\begin{gathered} x_{cm}=\frac{x_1{m}_1+x_2+m_2}{m_1+m_2} \\ {x}_{cm}=\frac{M\left(x_1\right)+\left(-m\right)x_2}{M+\left(-m\right)} \end{gathered}$$

Here, $x_1$is x-coordinate of center of mass of the whole plate and$x_2$ is the xcomponentof the centerof massof the addedplate. It can be seen that the x coordinate of center of mass of the whole plate is zero and x coordinate of center of mass of added mass is 2.0 m . Therefore,

$$\begin{gathered} x_{cm}=\frac{-m(2.0)}{9m-m} \\ {x}_{cm}=-0.25\mathrm{m} \end{gathered}$$

Hence,X coordinate of center of mass is,$x_{cm}=-0.25\mathrm{m}$

From the symmetry, the y componentof the centerof mass is found to be zero.

$∴y_{cm}=0\mathrm{m}$

Hence, Y coordinate of center of mass is, $∴y_{cm}=0\mathrm{m}$

Therefore, the relation between mass of whole plate and mass of small plate can be found using the center of mass of this small plate and center of mass of the entire plate, the center of mass of the resulting shape can be found.