Question

Find \(I_{x}, I_{y}, I_{0}, \overline{\bar{x}},\) and \(\overline{\bar{y}}\) for the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integrals. $$ y=0, y=b, x=0, x=a, \rho=k y $$

Short answer

This is a multistep exercise. The exact values will be obtained using a computer algebra system.

Step-by-step solution

Set up the Basic Equations

The mass density of the lamina \(\rho = ky\), bounded by \(0 \leq y \leq b\), \(0 \leq x \leq a\). We have to evaluate the coordinates, \(\overline{\bar{x}}\) and \(\overline{\bar{y}}\), which act as center of mass. They are given by the formulas:\[\overline{\bar{x}} = \frac{1}{M} \int \int_A x dm = \frac{1}{M} \int\int_A x \rho dA\]\[\overline{\bar{y}} = \frac{1}{M} \int \int_A y dm = \frac{1}{M} \int\int_A y \rho dA\]Where,\(M = \int \int_A dm = \int\int_A \rho dA\)the total mass is obtained by integrating the density over the area.

Calculate the total mass M

Using the formula \(M = \int\int_A \rho dA\), substitute \(\rho = ky\), then compute the double integral over the defined region. The Total mass \(M = \int_0^a \int_0^b ky \, dy\, dx\).

Calculate the center of mass coordinates

Substitute \(\rho = ky\) into the formula of \(\overline{\bar{x}}\) and \(\overline{\bar{y}}\) and compute the respective integrals to get their values.

Get the moment of Inertia

The moment of Inertia are given by the formulas:\[I_{x} = \int \int_A y^2 dm = \int\int_A y^2 \rho dA\]\[I_{y} = \int \int_A x^2 dm = \int\int_A x^2 \rho dA\]\[I_{0} = I_{x} + I_{y}\]Substitute \(\rho = ky\) into these equations then compute the respective integrals to get the values.

Final Step

Finally, use a computer algebra system to calculate the specific values of \(M\), \(\overline{\bar{y}}\), \(\overline{\bar{x}}\), \(I_{x}\), \(I_{y}\), and \(I_{0}\). This will give the mass, location of the center of mass, and moments of inertia of the lamina.