244.15.6.1 center of mass

In summary: First Moments}$$\displaystyle M_y=\int_{0}^{\sqrt{2}}\int_{2-x^2}^{1} \, dy\, dx=\int_{0}^{\sqrt{2}}\biggr[y\biggr]_{y=2-x^2}^{y=1}\, dx=\int_{0}^{\sqrt{2}}(x^2-1) \, dx=\int_{0}^{\sqrt{2}}\biggr[\frac{x^3}{3}-x\biggr]_0^{\
  • #1
karush
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$\textsf{Find the center of mass of a thin plate of density}$
$\textsf{ $\delta=3$ bounded by the lines $x=0, y=x$, and the parabola
$y=2-x^2$ in the $Q1$}$
$\begin{array}{llcr}\displaystyle
&\textit{Mass}\\
&&\displaystyle M=\iint\limits_{R}\delta \, dA\\
&\textit{First Moments}\\
&&\displaystyle M_y=\iint\limits_{R}x\delta \, dA
&\displaystyle M_x=\iint\limits_{R}y\delta \, dA\\
&\textit{Center of mass}\\
&&\displaystyle\bar{x}=\displaystyle\frac{M_y}{M},
\displaystyle\bar{y}=\displaystyle\frac{M_x}{M}\\
\\
&&\color{red}
{\displaystyle \, \bar{x}=\frac{5}{14},
\bar{y}=\displaystyle\frac{38}{35}}\\
\end{array}$ok I just barely had to time to post this
equations are just from reference
red is answer
 
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  • #2
Ok I'm starting over on this a small step at a time...

$\textsf{Find the center of mass of a thin plate of density}\\$
$\textsf{$\delta=3$ bounded by the lines $x=0, y=x$, and the parabola
$y=2-x^2$ in $Q1$}\\$
\begin{align*}\displaystyle
M&=\int_{0}^{\sqrt{2}}\int_{2-x^2}^{1}3 \, dy \, dx\\
&=3\int_{0}^{\sqrt{2}}\biggr[y\biggr]_{y=2-x^2}^{y=1}\, dx
=3\int_{0}^{\sqrt{2}}(x^2-1) \, dx
=3\biggr[\frac{x^3}{3}-x\biggr]_0^{\sqrt{2}}=\sqrt{2}\\
M_y&=\int_{0}^{\sqrt{2}}\int_{2-x^2}^{1} \, dy \, dx
=\int_{0}^{\sqrt{2}}\biggr[y\biggr]_{y=2-x^2}^{y=1}\, dx\\
&=\int_{0}^{\sqrt{2}}(x^2-1) \, dx = -\sqrt{2}\\
\end{align*}

something isn't happening right!


$\textsf{the answer utimately is:}\\$
$\color{red}{\, \bar{x}=\displaystyle\frac{5}{14},\bar{y}=\frac{38}{35}}$
 
Last edited:
  • #3
The first thing I would do is sketch the bounded area:

View attachment 7772

Now, let's compute the mass (noting that the curves $y=x$ and $y=2-x^2$ intersect at $x=1$ in QI):

\(\displaystyle m=\rho A=3\int_{0}^{1}\int_{x}^{2-x^2}\,dy\,dx=3\int_{0}^{1}2-x-x^2\,dx=3\left(2-\frac{1}{2}-\frac{1}{3}\right)=\frac{7}{2}\)

Next, let's compute the moments of the lamina:

\(\displaystyle M_x=3\int_{0}^{1}\int_{x}^{2-x^2}y\,dy\,dx=\frac{3}{2}\int_{0}^{1}\left(2-x^2\right)^2-x^2\,dx=\frac{3}{2}\int_{0}^{1}x^4-5x^2+4\,dx=\frac{19}{5}\)

\(\displaystyle M_y=3\int_{0}^{1}x\int_{x}^{2-x^2}\,dy\,dx=3\int_{0}^{1}2x-x^2-x^3\,dx=\frac{5}{4}\)

Hence:

\(\displaystyle \overline{x}=\frac{M_y}{m}=\frac{\frac{5}{4}}{\frac{7}{2}}=\frac{5}{14}\)

\(\displaystyle \overline{y}=\frac{M_x}{m}=\frac{\frac{19}{5}}{\frac{7}{2}}=\frac{38}{35}\)
 

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  • #4

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  • #5
karush said:
$\textsf{Find the center of mass of a thin plate of density}$
$\textsf{ $\delta=3$ bounded by the lines $\color{red}{x=0}$, $ y=x$, and the parabola
$y=2-x^2$ in the $Q1$}$

$\color{red}{x=0}$ is the y-axis as shown in Mark’s sketch, not the x-axis as shown in your sketch.
 

Related to 244.15.6.1 center of mass

1. What is the definition of "244.15.6.1 center of mass"?

The center of mass for an object is the point at which the mass of the object is evenly distributed in all directions. In the case of "244.15.6.1", this refers to a specific location or coordinate on an object or system.

2. How is the center of mass calculated?

The center of mass is calculated by taking into account the mass and position of each individual component of an object or system. It can be found using mathematical equations or through physical experiments.

3. What is the significance of the center of mass?

The center of mass is an important concept in physics and engineering. It helps to determine the stability, balance, and motion of an object or system. It is also useful in predicting the behavior of objects in different environments.

4. Can the center of mass be outside of an object?

Yes, the center of mass can be outside of an object. This can happen when the mass of an object is not evenly distributed. For example, a hammer's center of mass may be located outside of its physical body due to the heavier head compared to the handle.

5. How does the center of mass affect an object's motion?

The center of mass plays a crucial role in an object's motion. If an object is at rest, the center of mass will remain stationary. But if the object is in motion, the center of mass will move along the same path as the object. This is known as translational motion.

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