Problem 31 - Thomas Calculus

Question

Finding a moment of inertia and radius of gyration Find the moment of inertia and radius of gyration about the \(x\) -axis of a thin plate bounded by the parabola \(x=y-y^{2}\) and the line \(x+y=0\) if \(\delta(x, y)=x+y\)

Step-by-Step Solution

Answer

Find the moment of inertia integral and evaluate it; then calculate the radius of gyration with the obtained area.

Step 1: Identify the Region of Integration

First, determine the region bounded by the given equations: the parabola \(x = y - y^2\) and the line \(x + y = 0\). To find the intercepts and intersection points, solve these equations simultaneously.

Step 2: Solve for Intersection Points

Set \(x = y - y^2\) equal to \(x = -y\) and solve for \(y\):\[y - y^2 = -y\2y = y^2\y(y - 2) = 0\\]Thus, \(y = 0\) or \(y = 2\). Corresponding \(x\) values are \(x = 0\) at \(y = 0\) and \(x = -2\) at \(y = 2\). The region is bounded between these two extrema.

Step 3: Set up the Moments Equation

The moment of inertia \(I_x\) with respect to the \(x\)-axis for the thin plate is given by:\[I_x = \int_a^b \int_{c(y)}^{d(y)} y^2 \delta(x,y) \, dx \, dy\]Here, the density function is \(\delta(x,y) = x + y\). Determine the limits of \(x\) which range from \(x = -y\) (for the line) to \(x = y - y^2\) (for the parabola).

Step 4: Perform the Integration

Insert the specified density and simplify:\[I_x = \int_0^2 \int_{-y}^{y-y^2} y^2 (x + y) \, dx \, dy\]Calculate the inner integral first:\[\int_{-y}^{y-y^2} y^2 (x + y) \, dx = \int_{-y}^{y-y^2} (y^2 x + y^3) \, dx\]Evaluating this, compute the integral after substitution and integration by parts if necessary.

Step 5: Final Calculation of Integral

Evaluate the calculated expression from the previous step for the bounds of \(dx\) and integrate with respect to \(y\) from \(0\) to \(2\). This will provide the moment of inertia \(I_x\).

Step 6: Calculate Radius of Gyration

The radius of gyration \(k_x\) with respect to the \(x\)-axis is found using the formula:\[k_x = \sqrt{\frac{I_x}{A}}\]where \(A\) is the area of the region. First, find the area \(A\) by integrating the limits \(x = -y\) to \(x = y - y^2\).

Step 7: Evaluate \(A\) for the Radius of Gyration

Calculate the area \(A\) of the region:\[A = \int_0^2 \int_{-y}^{y-y^2} 1 \, dx \, dy\]Compute this definite integral and use it in the formula for radius of gyration \(k_x\).

Step 8: Compute the Radius of Gyration

Insert the computed area and moment of inertia into the equation from Step 6 to determine the radius of gyration:

Key Concepts

radius of gyration

The "radius of gyration" is a useful concept to consider when studying the distribution of a body's area or mass around its axis. It tells us how far from the axis the body's area or mass can be concentrated while still maintaining the same moment of inertia. This helps in understanding the stability and balance of objects.The formula used for the radius of gyration about the x-axis is:\[k_x = \sqrt{\frac{I_x}{A}}\]where:

• \(I_x\) is the moment of inertia with respect to the x-axis.
• \(A\) is the area of the region.

To calculate the radius of gyration, you first need the value of the moment of inertia and the area of the shape in question. Once both are determined, simply plug them into the formula. Remember, the radius of gyration provides an idea of how the mass is "spread out" in relation to the axis, indicating how it would "rotate or bend." Understanding this concept is critical in the design and analysis of structures and mechanical components.

integration region

The "integration region" in a problem like this refers to the area or volume over which an integral needs to be calculated. Here, it is the area enclosed by the parabola equation \(x = y - y^2\) and the line equation \(x + y = 0\).Understanding the integration region is crucial because it helps set up the correct bounds for definite integrals in the calculation process. To find this area, the intersection points of the two curves are first determined. In this example, by solving the equations

• \(y - y^2 = -y\)
• \(y(y - 2) = 0\)
• giving \(y = 0\) or \(y = 2\)

This means the curves intersect at \((0, 0)\) and \((2, -2)\), creating the limits for the integration function. The region over which we integrate is from \(x = -y\) to \(x = y - y^2\), bounded between \(y = 0\) and \(y = 2\). This clearly defined region is essential for correctly calculating both the area and the moment of inertia.

density function

A "density function" describes how mass or another property is distributed over a region. In our problem, the density function is given by \(\delta(x, y) = x + y\). This means the density depends on the specific location within the plate.The density function is a critical component of evaluating integrals for calculating the moment of inertia, as it can significantly influence the result.For the plate in question, the moment of inertia integral was set up as:\[I_x = \int_0^2 \int_{-y}^{y-y^2} y^2 (x + y) \, dx \, dy\]Here, \(x + y\) is our density function, and it is multiplied by \(y^2\) in the integrand. This reflects the role density plays in the physical trait being measured—in this case, rotational inertia.Understanding the density function aids in comprehending how various regions of the plate contribute differently to the overall inertia, depending on their position and given the varying densities described by \(x + y\).

moment of inertia formula

The "moment of inertia formula" is used to calculate the inertia of an object like our thin plate relative to a specified axis. It is a measure of how difficult it is to change the object's rotational motion about that axis.In our context, the moment of inertia \(I_x\) is calculated using the formula:\[I_x = \int_a^b \int_{c(y)}^{d(y)} y^2 \delta(x,y) \, dx \, dy\]This formula integrates across the defined region of the plate, from limits \(c(y)\) to \(d(y)\), incorporating the density \(\delta(x,y)\) into the calculation. Breaking it down:

• \(y^2\) represents perpendicular distance squared from the axis, capturing rotational effects.
• The entire expression \(y^2 \delta(x,y)\) inside the integral denotes how local regions contribute to overall inertia.

In simpler terms, this formula considers not just the distribution of area/mass but how each point's distribution and density affect the resistance to rotational change. Knowing how to employ the moment of inertia formula is pivotal in fields like engineering and physics, as it is key to analyzing mechanical systems and structures.