Question

Figure 9-82 shows a uniform square plate of edge length 6d=6.0 m from which a square piece of edge length 2dhas been removed. What are (a) the xcoordinate and (b) the ycoordinate of the center of mass of the remaining piece?

Short answer

1. X coordinate of center of mass is,$x_{cm}=-0.25\mathrm{m}$
2. Y coordinate of center of mass is,role="math" localid="1661252999190" $y_{cm}=0\mathrm{m}$

Step-by-step solution

Step 1: Given

$6d=6m$

Determining the concept

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.

(a) Determining if length  is removed then what is the x coordinate of the center of mass

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

$Mass=density\times volume$

$M=pAL$

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,$x_{cm}=-0.25\mathrm{m}$

(b) Determining if length of  is removed then what is y coordinate of the center of mass

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

$\therefore y_{cm}=0\mathrm{m}$

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

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.