Chapter 7.1, Problem 45E

a.

To determine

To show: The x-coordinate of the center of mass of a thin plate is $\overline{x}=\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}}$ , where $m=\delta dA$ , $\delta$ is the density and $A$ is the area of the region.

Explanation of Solution

Given information:

The region is cut into strips parallel to y-axis.

Calculation:

Let $m_k$ be the mass of the strip at $x=x_k$ , and $\Delta A$ be the area of that strip. Then, the mass of the strip is $m_k=\delta \Delta A$ .

The mass of an element strip of area $\Delta A$ will be $\Delta m=\delta \Delta A$ .

The moment of all the masses about y-axis is $M_y={\displaystyle \sum m_k{x}_k}$ and the total mass of the region is $M={\displaystyle \sum m_k}$ .

Then, the x-coordinate of the center of mass is $\overline{x}=\frac{{\displaystyle \sum m_k{x}_k}}{{\displaystyle \sum m_k}}$ or $\overline{x}=\frac{{\displaystyle \sum \delta \Delta A{x}_k}}{{\displaystyle \sum \delta \Delta A}}$ .

Since $\delta \Delta A=\Delta m$ , the x-coordinate of the center of mass is $\overline{x}=\frac{{\displaystyle \sum x_k\Delta m}}{{\displaystyle \sum \Delta m}}$ .

Now, take the limit of sum to get $\overline{x}=\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}}$ .

Conclusion:

The x-coordinate of the center of mass is $\overline{x}=\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}}$ .

b.

To determine

To show: The y-coordinate of the center of mass of a thin plate is $\overline{y}=\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}$ , where $m=\delta dA$ , $\delta$ is the density and $A$ is the area of the region.

Explanation of Solution

Given information:

The region is cut into strips parallel to y-axis.

Calculation:

Let $m_k$ be the mass of the strip at $y=y_k$ , and $\Delta A$ be the area of that strip. Then, the mass of the strip is $m_k=\delta \Delta A$ .

The mass of an element strip of area $\Delta A$ will be $\Delta m=\delta \Delta A$ .

The moment of all the masses about x-axis is $M_x={\displaystyle \sum m_k{y}_k}$ and the total mass of the region is $M={\displaystyle \sum m_k}$ .

Then, the x-coordinate of the center of mass is $\overline{y}=\frac{{\displaystyle \sum m_k{y}_k}}{{\displaystyle \sum m_k}}$ or $\overline{y}=\frac{{\displaystyle \sum \delta \Delta A{y}_k}}{{\displaystyle \sum \delta \Delta A}}$ .

Since $\delta \Delta A=\Delta m$ , the x-coordinate of the center of mass is $\overline{y}=\frac{{\displaystyle \sum y_k\Delta m}}{{\displaystyle \sum \Delta m}}$ .

Now, take the limit of sum to get $\overline{y}=\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}$ .

Conclusion:

The x-coordinate of the center of mass is $\overline{y}=\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}$ .