# 数学代写|微积分代写Calculus代写|Masses Distributed over a Plane Region

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## 数学代写|微积分代写Calculus代写|Masses Distributed over a Plane Region

Suppose that we have a finite collection of masses located in the plane, with mass $m_k$ at the point $\left(x_k, y_k\right)$ (see Figure 6.46). The mass of the system is
System mass: $\quad M=\sum m_k$.
Each mass $m_k$ has a moment about each axis. Its moment about the $x$-axis is $m_k y_k$, and its moment about the $y$-axis is $m_k x_k$. The moments of the entire system about the two axes are
$$\begin{array}{ll} \text { Moment about } x \text {-axis: } & M_x=\sum m_k y_k, \ \text { Moment about } y \text {-axis: } & M_y=\sum m_k x_k . \end{array}$$
The $x$-coordinate of the system’s center of mass is defined to be
$$\bar{x}=\frac{M_y}{M}=\frac{\sum m_k x_k}{\sum m_k} .$$
With this choice of $\bar{x}$, as in the one-dimensional case, the system balances about the line $x=\bar{x}$ (Figure 6.47).
The $y$-coordinate of the system’s center of mass is defined to be
$$\bar{y}=\frac{M_x}{M}=\frac{\sum m_k y_k}{\sum m_k} .$$
With this choice of $\bar{y}$, the system balances about the line $y=\bar{y}$ as well. The torques exerted by the masses about the line $y=\bar{y}$ cancel out. Thus, as far as balance is concerned, the system behaves as if all its mass were at the single point $(\bar{x}, \bar{y})$. We call this point the system’s center of mass.

## 数学代写|微积分代写Calculus代写|Thin, Flat Plates

In many applications, we need to find the center of mass of a thin, flat plate: a disk of aluminum, say, or a triangular sheet of steel. In such cases, we assume the distribution of mass to be continuous, and the formulas we use to calculate $\bar{x}$ and $\bar{y}$ contain integrals instead of finite sums. The integrals arise in the following way.

Imagine that the plate occupying a region in the $x y$-plane is cut into thin strips parallel to one of the axes (in Figure 6.48 , the $y$-axis). The center of mass of a typical strip is $(\tilde{x}, \tilde{y})$. We treat the strip’s mass $\Delta m$ as if it were concentrated at $(\tilde{x}, \tilde{y})$. The moment of the strip about the $y$-axis is then $\tilde{x} \Delta m$. The moment of the strip about the $x$-axis is $\tilde{y} \Delta m$. Equations (3) and (4) then become
$$\bar{x}=\frac{M_y}{M}=\frac{\sum \tilde{x} \Delta m}{\sum \Delta m}, \quad \bar{y}=\frac{M_x}{M}=\frac{\sum \tilde{y} \Delta m}{\sum \Delta m} .$$
These sums are Riemann sums for integrals, and they approach these integrals in the limit as the strips become narrower and narrower. We write these integrals symbolically as
$$\bar{x}=\frac{\int \tilde{x} d m}{\int d m} \quad \text { and } \quad \bar{y}=\frac{\int \tilde{y} d m}{\int d m} .$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Masses Distributed over a Plane Region

$$\begin{array}{ll} \text { Moment about } x \text {-axis: } & M_x=\sum m_k y_k, \ \text { Moment about } y \text {-axis: } & M_y=\sum m_k x_k . \end{array}$$

$$\bar{x}=\frac{M_y}{M}=\frac{\sum m_k x_k}{\sum m_k} .$$

$$\bar{y}=\frac{M_x}{M}=\frac{\sum m_k y_k}{\sum m_k} .$$

## 数学代写|微积分代写Calculus代写|Thin, Flat Plates

$$\bar{x}=\frac{M_y}{M}=\frac{\sum \tilde{x} \Delta m}{\sum \Delta m}, \quad \bar{y}=\frac{M_x}{M}=\frac{\sum \tilde{y} \Delta m}{\sum \Delta m} .$$

$$\bar{x}=\frac{\int \tilde{x} d m}{\int d m} \quad \text { and } \quad \bar{y}=\frac{\int \tilde{y} d m}{\int d m} .$$

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