Answer:
4) y = -1/2x + 1;
Step-by-step explanation:
maybe
or maybe not
The equation of the line through (0, 1) and (<em>c</em>, 0) is
<em>y</em> - 0 = (0 - 1)/(<em>c</em> - 0) (<em>x</em> - <em>c</em>) ==> <em>y</em> = 1 - <em>x</em>/<em>c</em>
Let <em>L</em> denote the given lamina,
<em>L</em> = {(<em>x</em>, <em>y</em>) : 0 ≤ <em>x</em> ≤ <em>c</em> and 0 ≤ <em>y</em> ≤ 1 - <em>x</em>/<em>c</em>}
Then the center of mass of <em>L</em> is the point
with coordinates given by
![\bar x = \dfrac{M_x}m \text{ and } \bar y = \dfrac{M_y}m](https://tex.z-dn.net/?f=%5Cbar%20x%20%3D%20%5Cdfrac%7BM_x%7Dm%20%5Ctext%7B%20and%20%7D%20%5Cbar%20y%20%3D%20%5Cdfrac%7BM_y%7Dm)
where
is the first moment of <em>L</em> about the <em>x</em>-axis,
is the first moment about the <em>y</em>-axis, and <em>m</em> is the mass of <em>L</em>. We only care about the <em>y</em>-coordinate, of course.
Let <em>ρ</em> be the mass density of <em>L</em>. Then <em>L</em> has a mass of
![\displaystyle m = \iint_L \rho \,\mathrm dA = \rho\int_0^c\int_0^{1-\frac xc}\mathrm dy\,\mathrm dx = \frac{\rho c}2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20m%20%3D%20%5Ciint_L%20%5Crho%20%5C%2C%5Cmathrm%20dA%20%3D%20%5Crho%5Cint_0%5Ec%5Cint_0%5E%7B1-%5Cfrac%20xc%7D%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%20%3D%20%5Cfrac%7B%5Crho%20c%7D2)
Now we compute the first moment about the <em>y</em>-axis:
![\displaystyle M_y = \iint_L x\rho\,\mathrm dA = \rho \int_0^c\int_0^{1-\frac xc}x\,\mathrm dy\,\mathrm dx = \frac{\rho c^2}6](https://tex.z-dn.net/?f=%5Cdisplaystyle%20M_y%20%3D%20%5Ciint_L%20x%5Crho%5C%2C%5Cmathrm%20dA%20%3D%20%5Crho%20%5Cint_0%5Ec%5Cint_0%5E%7B1-%5Cfrac%20xc%7Dx%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%20%3D%20%5Cfrac%7B%5Crho%20c%5E2%7D6)
Then
![\bar y = \dfrac{M_y}m = \dfrac{\dfrac{\rho c^2}6}{\dfrac{\rho c}2} = \dfrac c3](https://tex.z-dn.net/?f=%5Cbar%20y%20%3D%20%5Cdfrac%7BM_y%7Dm%20%3D%20%5Cdfrac%7B%5Cdfrac%7B%5Crho%20c%5E2%7D6%7D%7B%5Cdfrac%7B%5Crho%20c%7D2%7D%20%3D%20%5Cdfrac%20c3)
but this clearly isn't independent of <em>c</em> ...
Maybe the <em>x</em>-coordinate was intended? Because we would have had
![\displaystyle M_x = \iint_L y\rho\,\mathrm dA = \rho \int_0^c\int_0^{1-\frac xc}y\,\mathrm dy\,\mathrm dx = \frac{\rho c}6](https://tex.z-dn.net/?f=%5Cdisplaystyle%20M_x%20%3D%20%5Ciint_L%20y%5Crho%5C%2C%5Cmathrm%20dA%20%3D%20%5Crho%20%5Cint_0%5Ec%5Cint_0%5E%7B1-%5Cfrac%20xc%7Dy%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%20%3D%20%5Cfrac%7B%5Crho%20c%7D6)
and we get
![\bar x = \dfrac{M_x}m = \dfrac{\dfrac{\rho c}6}{\dfrac{\rho c}2} = \dfrac13](https://tex.z-dn.net/?f=%5Cbar%20x%20%3D%20%5Cdfrac%7BM_x%7Dm%20%3D%20%5Cdfrac%7B%5Cdfrac%7B%5Crho%20c%7D6%7D%7B%5Cdfrac%7B%5Crho%20c%7D2%7D%20%3D%20%5Cdfrac13)
Answer:1/10
1/100
Step-by-step explanation:
Answer:
1.9a
Step-by-step explanation:
6.7a - 4.8a = 1.9a
Answer:
force of gravity
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