Notes: `y= 1/2x^2 , y=0, x=2` Find the x and y moments of inertia and center of mass for the laminas of uniform density `p` bounded by the graphs of...

Wednesday, 18 September 2013

$\displaystyle{y}=\frac{{1}}{{2}}{{x}}^{{2}},{y}={0},{x}={2}$ Find the x and y moments of inertia and center of mass for the laminas of uniform density $\displaystyle{p}$ bounded by the graphs of...

The center of Mass is:

$\displaystyle{\left({x}_{{{c}{m}}},{y}_{{{c}{m}}}\right)}={\left(\frac{{M}_{{y}}}{{M}},\frac{{M}_{{x}}}{{M}}\right)}$

Where the moments of mass are defined as:

$\displaystyle{M}_{{x}}=\int\int_{{A}}\rho{\left({x},{y}\right)}\cdot{y}{\left.{d}{y}\right.}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\int\int_{{A}}\rho{\left({x},{y}\right)}\cdot{x}{\left.{d}{y}\right.}{\left.{d}{x}\right.}$

The total mass is defined as:

$\displaystyle{M}=\int\int_{{A}}\rho{\left({x},{y}\right)}{\left.{d}{y}\right.}{\left.{d}{x}\right.}$

First, lets find the total mass.

$\displaystyle{M}={\int}^{{2}}_{0}{\left[{\int}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\rho{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}=\rho{\int}^{{2}}_{0}{\left[{y}{{\mid}}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}=\rho{\int}^{{2}}_{0}\frac{{1}}{{2}}{{x}}^{{2}}{\left.{d}{x}\right.}$

$\displaystyle{M}=\frac{\rho}{{2}}{\left(\frac{{1}}{{3}}\right)}{{x}}^{{3}}{{\mid}}^{{2}}_{0}$

$\displaystyle{M}=\frac{\rho}{{2}}{\left(\frac{{1}}{{3}}\right)}{{2}}^{{3}}$

$\displaystyle{M}=\frac{{4}}{{3}}\rho$

Now lets find the x moment of mass.

$\displaystyle{M}_{{x}}={\int}^{{2}}_{0}{\left[{\int}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\rho\cdot{y}{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{x}}=\rho{\int}^{{2}}_{0}{\left[{\left(\frac{{1}}{{2}}\right)}{{y}}^{{2}}{{\mid}}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{x}}=\rho{\int}^{{2}}_{0}{\left[{\left(\frac{{1}}{{2}}\right)}{{\left(\frac{{1}}{{2}}{{x}}^{{2}}\right)}}^{{2}}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{x}}=\rho{\int}^{{2}}_{0}{\left(\frac{{1}}{{8}}\right)}{{x}}^{{4}}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{x}}={\left(\frac{\rho}{{8}}\right)}{\left(\frac{{1}}{{5}}\right)}{{x}}^{{5}}{{\mid}}^{{2}}_{0}$

$\displaystyle{M}_{{x}}=\frac{\rho}{{40}}\cdot{{2}}^{{5}}=\frac{{32}}{{40}}\rho=\frac{{4}}{{5}}\rho$

Now the y moment of mass.

$\displaystyle{M}_{{y}}={\int}^{{2}}_{0}{\left[{\int}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\rho\cdot{x}{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\rho{\int}^{{2}}_{0}{x}{\left[{\int}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}{\left.{d}{y}\right.}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\rho{\int}^{{2}}_{0}{x}{\left[{y}{{\mid}}^{{\frac{{1}}{{2}}{{x}}^{{2}}}}_{0}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\rho{\int}^{{2}}_{0}{x}{\left[\frac{{1}}{{2}}{{x}}^{{2}}\right]}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\frac{\rho}{{2}}{\int}^{{2}}_{0}{{x}}^{{3}}{\left.{d}{x}\right.}$

$\displaystyle{M}_{{y}}=\frac{\rho}{{2}}{\left(\frac{{1}}{{4}}\right)}{{x}}^{{4}}{{\mid}}^{{2}}_{0}$

$\displaystyle{M}_{{y}}=\frac{\rho}{{2}}{\left(\frac{{1}}{{4}}\right)}{{2}}^{{4}}={2}\rho$

Therefore the center of mass is:

$\displaystyle{\left({x}_{{{c}{m}}},{y}_{{{c}{m}}}\right)}={\left(\frac{{M}_{{y}}}{{M}},\frac{{M}_{{x}}}{{M}}\right)}={\left(\frac{{{2}\rho}}{{\frac{{4}}{{3}}\rho}},\frac{{\frac{{4}}{{5}}\rho}}{{\frac{{4}}{{3}}\rho}}\right)}={\left(\frac{{3}}{{2}},\frac{{3}}{{5}}\right)}$

The moments of inerita or the second moments of the lamina are:

$\displaystyle{I}_{{x}}=\int\int_{{A}}\rho{\left({x},{y}\right)}\cdot{{y}}^{{2}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}$

$\displaystyle{I}_{{y}}=\int\int_{{A}}\rho{\left({x},{y}\right)}\cdot{{x}}^{{2}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}$

I won't solve these integrals step by step since they are very similar to the others, but you will find that:

$\displaystyle{I}_{{x}}=\frac{{16}}{{21}}\rho$

$\displaystyle{I}_{{y}}=\frac{{16}}{{5}}\rho$

Notes

Wednesday, 18 September 2013

$\displaystyle{y}=\frac{{1}}{{2}}{{x}}^{{2}},{y}={0},{x}={2}$ Find the x and y moments of inertia and center of mass for the laminas of uniform density $\displaystyle{p}$ bounded by the graphs of...

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Wednesday, 18 September 2013

Find the x and y moments of inertia and center of mass for the laminas of uniform density bounded by the graphs of...

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle{y}=\frac{{1}}{{2}}{{x}}^{{2}},{y}={0},{x}={2}$ Find the x and y moments of inertia and center of mass for the laminas of uniform density $\displaystyle{p}$ bounded by the graphs of...

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?