Problem 20 - Thomas Calculus

Question

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. Consider a thin plate of constant density \(\delta\) lies in the region bounded by the graphs of \(y=\sqrt{x}\) and \(x=2 y .\) Find the plate's \begin{equation}\begin{array}{l}{\text { a. moment about the } x \text { -axis. }} \\ {\text { b. moment about the } y \text { -axis. }} \\ {\text { c. moment about the line } x=5 \text { . }} \\ {\text { d. moment about the line } x=-1} \\ {\text { e. moment about the line } y=2 \text { . }} \\\ {\text { f. moment about the line } y=-3 \text { . }} \\ {\text { g. mass. }} \\\ {\text { h. center of mass. }}\end{array}\end{equation}

Step-by-Step Solution

Answer

The center of mass is computed using the specific integrals for moments and mass.

Step 1: Find Intersection Points

Equate the equations for the curves to find intersection points. Solve \(y=\sqrt{x}\) and \(x=2y\) simultaneously:\[\sqrt{x}=y, \text{ and } x=2y\] implies \[\sqrt{x}=\frac{x}{2}\] to simplify\[2\sqrt{x}=x\] implies \[x=0,4\]. Thus, the region is bounded between \(x=0\) and \(x=4\).

Step 2: Set up Limits for Integration

Since \(x=2y\), we find \(y=\frac{x}{2}\). And from \(y=\sqrt{x}\), we know \(y\) varies over \(\frac{x}{2}\) to \(\sqrt{x}\) from \(x=0\) to \(x=4\).

Step 3: Moment about the x-axis

The moment about the x-axis \( M_x \) is calculated using the formula: \[M_x = \delta \int_{0}^{4} \left(\frac{x}{2} + \sqrt{x}\right) \frac{1}{2} \Delta y \ dx\]. Inserting the expressions for \(y\) and simplifying, calculate the integral.

Step 4: Moment about the y-axis

The moment about the y-axis \( M_y \) is calculated using the formula: \[M_y = \delta \int_{0}^{4} x\left(\sqrt{x} - \frac{x}{2}\right)\ dx\]. Calculate this integral by substituting the expressions of the functions.

Step 5: Moments about the line x=5 and x=-1

Compute the moment about line \(x=5\):\[M_{x=5} = \delta \int_{0}^{4} (5 - x)\left(\sqrt{x} - \frac{x}{2}\right)\ dx\]. Similarly, for line \(x=-1\):\[M_{x=-1} = \delta \int_{0}^{4} (x+1)\left(\sqrt{x} - \frac{x}{2}\right)\ dx\]. Evaluate these integrals.

Step 6: Moments about the line y=2 and y=-3

Compute the moments \( M_{y=2} \) and \( M_{y=-3} \):\[M_{y=2} = \delta \int_{0}^{4} \left(\frac{x}{2} + 2\right) \left(\sqrt{x} - \frac{x}{2}\right)\ dx\] and\[M_{y=-3} = \delta \int_{0}^{4} \left(\frac{x}{2} + 3\right)\left(\sqrt{x} - \frac{x}{2}\right)\ dx\]. Evaluate these integrals.

Step 7: Calculate the Mass

Calculate the mass \(m\) of the plate using \[m = \delta \int_{0}^{4} \left(\sqrt{x} - \frac{x}{2}\right)\ dx\]. Solve for \(m\).

Step 8: Calculate the Center of Mass

The coordinates of the center of mass \((\overline{x}, \overline{y})\) are given by \(\overline{x} = \frac{M_y}{m}\) and \(\overline{y} = \frac{M_x}{m}\). Substitute in the computed values of moments and mass to find the center of mass.

Key Concepts

Moment About the X-Axis

The moment about the x-axis is an essential part of understanding how the mass of an object is distributed with respect to this axis. For a thin plate lying on a region defined by two curves, we calculate the moment by integrating the product of the density of the plate, the y-coordinate of the mass elements, and the differential element of area.

In this specific case, the moment about the x-axis, denoted as \( M_x \), utilizes the formula \[ M_x = \delta \int_{0}^{4} \left(\frac{x}{2} + \sqrt{x}\right) \frac{1}{2} \, dx \].

Here, \(\delta\) represents the constant density of the plate.

The integral bounds are from 0 to 4, the limits determined by the region's intersection points.

The formula involves the sum of two functions, one derived from each bounding curve, over the specified interval. The integration provides a measure of how the plate's mass is distributed vertically about the x-axis, thus allowing us to determine the center of mass along this axis.

Moment About the Y-Axis

To find how mass is distributed with respect to the y-axis, we calculate the moment about the y-axis, \( M_y \).
The moment about the y-axis considers how far mass elements are from the y-axis. This helps in determining the horizontal distribution of the object.

In our scenario, the calculation is given by:\[ M_y = \delta \int_{0}^{4} x\left(\sqrt{x} - \frac{x}{2}\right) \, dx \].This integral contains:

• The constant \(\delta \), representing the constant density of the plate.
• \( x \), which indicates each location's distance from the y-axis.
• The expression \( \sqrt{x} - \frac{x}{2} \) as the depth of the plate defined by the boundary functions.

The result of the integration process gives \( M_y \), the initial step towards locating the center of mass horizontally. By completing this integral, we detail the y-axis's impact on the mass distribution across the thin plate.

Integration

In calculating moments and mass in physics, integration is a powerful tool. It allows us to handle continually varying quantities, like density or position, over a defined region.
Through integration, accumulative effects such as the total mass or moment (turning effect) of the plate are summed.

For example, the integration range taken from 0 to 4 is based on the intersection points of the given curves,\[ \int_{0}^{4} \text{functions of } x \, dx \].
During these problems, integration generally follows these steps:

• Determine the limits of integration, based on where the curves intersect and define the region of interest.
• Create integrands from the expressions of the curves, incorporating density and position.
• Evaluate the integral to find moments or mass.

Utilizing integration helps us translate a geometric and physical problem into a solvable mathematical format, allowing us to determine properties like mass and the center of mass.

Thin Plate

A thin plate in mathematics and physics refers to an object that has one dimension significantly smaller than the other two, making it effectively two-dimensional. Understanding this simplicity allows concentration on the mass distribution across the defined plane.

In this exercise, we're dealing with a thin plate of constant density in a region bounded by curves, where the plate's surface defines the problem area.

• The "thin" description assures that every small element of the plate is uniform in thickness, which simplifies calculations since density only needs to be considered over two dimensions.
• It forms the basis for calculating moments and centers of mass because the third dimension's effects can be neglected.

Because the plate remains uniform, computations involve only the surface densities and the areas, simplifying our mathematical approach.

Constant Density

Constant density means every unit of area on our thin plate has the same mass.
This uniformity makes calculations easier since density \(\delta\) remains unchanged across the entire region.

Having a constant density simplifies integration greatly:

• It lets us factor out density \(\delta\) from the integrals used in calculating moments and mass.
• As a result, the expressions primarily depend on the geometry defined by the boundary curves.

Ultimately, constant density provides a straightforward way to explore how mass is distributed in the plate, as it eliminates variables and focuses solely on spatial factors. It's essential for understanding "linear" approaches that rely primarily on area and boundary geometries.