Textbook Problem

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Finding Moments of Inertia and Radii of Gyration

In Exercises 35–38, find $I_x,I_y,I_0,\overline{\stackrel{¯}{x}}$ and $\overline{\stackrel{¯}{y}}$ for the lamina bounded by the graphs of the equations.

$y=x,y=x^2,\rho =kxy$

To determine

To Calculate: The Moments of Inertia and Radii of Gyration of the provided data..

Explanation of Solution

Given:

The given equation is,

$\begin{array}{l}y=x,y=x^2,\\ \rho =kxy\end{array}$

Formula Used:

The formula is,

$m={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}\rho (x,y)dydx}}$

$M_x={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}y\cdot \rho (x,y)dydx}}$

$M_y={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}x\cdot \rho (x,y)dydx}}$

$\left(\overline{x},\overline{y}\right)=\left(\frac{M_y}{m},\frac{M_x}{m}\right)$

$I_x={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}{y}^2\cdot \rho (x,y)dydx}}$

$I_y={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}{x}^2\cdot \rho (x,y)dydx}}$

$I_0={\displaystyle {\int }_{x_1}^{x_2}{\displaystyle {\int }_{y_1}^{y_2}(x^2+y^2)\cdot \rho (x,y)dydx}}$

$\begin{array}{l}\overline{\overline{x}}=\sqrt{\frac{I_y}{m}}\\ \overline{\overline{y}}=\sqrt{\frac{I_x}{m}}\end{array}$

Where $\left(x_1,x_2\right)$ and $\left(y_1,y_2\right)$ are the limits of the integration.

Calculation:

To calculate the moment of inertia and radii of gyrationdraw the graph for the given equations by performing the following steps that is,

1. First Draw the reference axis in xy coordinates.

2. then Plot the given equations:

$\begin{array}{l}y=x,\\ y=x^2\end{array}$

3. Mark the points on the plot and then connect the points.

4. The area of interest is the area inscribed by the equations, so shade the inside area of the graph.

5. The final plot is as shown below: