Chapter 8.1, Problem 39E

To determine

Tofind the centre of mass.

Answer to Problem 39E

The centre of the mass is found to be $\left(\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}},\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}\right)$ .

Explanation of Solution

Given information:

Suppose we have a finite collection of masses in the co-ordinate plane, the mass $m_k$ located at the point $\left(x_k,y_k\right)$ as shown in the figure.

  

From given each mass $m_k$ has moment $m_k{y}_k$ with respect to the x-axis and moment $m_k{x}_k$ about the y-axis.

The moment about the entire system with respect to the two axes are:

Moment about the x-axis $M_x={\displaystyle \sum m_k{y}_k}$ ,

Moment about the y-axis $M_y={\displaystyle \sum m_k{x}_k}$ ,

The centre of mass is $\left(\overline{x},\overline{y}\right)$ where,

  $\begin{array}{l}\overline{x}=\frac{M_y}{M}=\frac{{\displaystyle \sum m_k{x}_k}}{{\displaystyle \sum m_k}}\\ \overline{y}=\frac{M_x}{M}=\frac{{\displaystyle \sum m_k{x}_k}}{{\displaystyle \sum m_k}}\end{array}$

Substitute $dm=\delta dA$ replace for $m_k$ , let $dA\to 0$ and $k\to \infty$ in the centre of mass equation.

Then,

  $\underset{k\to \infty ,dA\to 0}{\lim }\left(\frac{{\displaystyle \sum x_k\delta dA}}{{\displaystyle \sum m_k}}\right)=\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}}$

Substitute $dm=\delta dA$ replace for $m_k$ , let $dA\to 0$ and $k\to \infty$ in the centre of mass equation along y-axis.

Then,

  $\underset{k\to \infty ,dA\to 0}{\lim }\left(\frac{{\displaystyle \sum y_k\delta dA}}{{\displaystyle \sum m_k}}\right)=\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}$

Therefore, the centre of the mass is found to be $\left(\frac{{\displaystyle \int xdm}}{{\displaystyle \int dm}},\frac{{\displaystyle \int ydm}}{{\displaystyle \int dm}}\right)$ .