232.15.6.31Find the center of mass

[karush]

$\tiny{232.15.6.31}$
$\textsf{Find the center of mass of the following solid,
assuming a constant density.}$
$\textsf{Sketch the region indicate the location of the centriods}$
$\textsf{Use symetry when possible and choose a convient }$
$\textsf{coordinate system the sliced solid cylinder bounded}$
$\textsf{by $x^2 +y^2=196$, $z=0$ and $y+z=1$}$

ok I won't deal with this till morning
but if suggestions
never done mass before

 

[HallsofIvy]

First, the "center of mass assuming constant density" is the "centroid" of a figure. "Centroid" is a mathematics term while "center of mass" is a physics term. Second, your use of the plural "centroids" is peculiar. A geometric figure only has one centroid.

Given a three dimensional object, S, the volume is \(\displaystyle V= \int\int_S\int dV\). The coordinates of the centroid are then \(\displaystyle \overline{x}= \frac{\int\int_S\int xdV}{V}\), \(\displaystyle \overline{y}= \frac{\int\int_S\int ydV}{V}\), and \(\displaystyle \overline{z}= \frac{\int\int_S\int zdV}{V}\). Since the figure here is a cylinder I would use cylindrical coordinates, r, \(\displaystyle \theta\), and z. To find V, integrate dV with r from 0 to 13, \(\displaystyle \theta\) from 0 to \(\displaystyle 2\pi\) and z from 0 to \(\displaystyle 1- y= 1- r sin(\theta)\).

 

[karush]

$\tiny{232.15.6.31}$
$\textsf{Find the center of mass of the following solid,
assuming a constant density.}\\
\textsf{a. Sketch the region indicate the location of the centriod}\\
\textsf{b. Use symmetry when possible}\\
\textsf{choose a convenient coordinate system
the sliced solid cylinder bounded by}$
\begin{align*}\displaystyle
x^2 +y^2&=196\\
z&=0\\
y+z&=1
\end{align*}

ok the radius of the circle was $7$
and presume the slant is $45^o$ so
centroid is $0,0,?$View attachment 7351

 

[karush]

First, the "center of mass assuming constant density" is the "centroid" of a figure. "Centroid" is a mathematics term while "center of mass" is a physics term. Second, your use of the plural "centroids" is peculiar. A geometric figure only has one centroid.

Given a three dimensional object, S, the volume is \(\displaystyle V= \int\int_S\int dV\). The coordinates of the centroid are then \(\displaystyle \overline{x}= \frac{\int\int_S\int xdV}{V}\), \(\displaystyle \overline{y}= \frac{\int\int_S\int ydV}{V}\), and \(\displaystyle \overline{z}= \frac{\int\int_S\int zdV}{V}\). Since the figure here is a cylinder I would use cylindrical coordinates, r, \(\displaystyle \theta\), and z. To find V, integrate dV with r from 0 to 13, \(\displaystyle \theta\) from 0 to \(\displaystyle 2\pi\) and z from 0 to \(\displaystyle 1- y= 1- r sin(\theta)\).

where do you get $13$ the $\sqrt{196}=14$ ?

 

[HallsofIvy]

where do you get $13$ the $\sqrt{196}=14$ ?

My eyes are going wonky: 13 is the square root of 169.