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Pavel [41]
2 years ago
7

A triangular lamina has vertices (0, 0), (0, 1) and (c, 0) for some positive constant c. Assuming constant mass density, show th

at the y-coordinate of the center of mass of the lamina is independent of the constant c.
Mathematics
1 answer:
a_sh-v [17]2 years ago
8 0

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 (\bar x,\bar y) with coordinates given by

\bar x = \dfrac{M_x}m \text{ and } \bar y = \dfrac{M_y}m

where M_x is the first moment of <em>L</em> about the <em>x</em>-axis, M_y 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

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

Then

\bar y = \dfrac{M_y}m = \dfrac{\dfrac{\rho c^2}6}{\dfrac{\rho c}2} = \dfrac c3

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

and we get

\bar x = \dfrac{M_x}m = \dfrac{\dfrac{\rho c}6}{\dfrac{\rho c}2} = \dfrac13

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