Problem 45 - Calculus Single Variable

Question

Find the center of mass of the given region \(\mathcal{R},\) assuming that it has uniform unit mass density. \(\mathcal{R}\) is the region bounded above by \(y=1\) for \(-1 \leq x \leq 0\) and by \(y=1-x\) for \(0 \leq x \leq 1\), below by the \(x\) -axis, and on the left by \(x=-1\).

Step-by-Step Solution

Answer

The center of mass is \((-\frac{1}{3}, \frac{1}{2})\).

Step 1: Understand the Problem

We need to find the center of mass of a region \(\mathcal{R}\) with uniform density, bounded by given lines. Region \(\mathcal{R}\) is a combination of a rectangle and a triangle, bounded at the top by \(y=1\) for \(-1 \leq x \leq 0\) and by \(y=1-x\) for \(0 \leq x \leq 1\), below by the x-axis, left by \(x=-1\).

Step 2: Divide the Region

The region \(\mathcal{R}\) can be split into two parts: \(R_1\), a rectangle from \((-1, 0)\) to \((0, 1)\), and \(R_2\), a right triangle with vertices at \((0, 0)\), \((1, 0)\), and \((0, 1)\).

Step 3: Calculate Area of Each Region

The area of \(R_1\) is calculated by multiplying its base and height: \(A_1 = 1 \times 1 = 1\). The area of \(R_2\) is \(A_2 = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}\).

Step 4: Find Center of Mass of Each Region

For the rectangle \(R_1\), the center of mass is \((x_1, y_1) = (-0.5, 0.5)\). For the triangle \(R_2\), the center of mass is \((x_2, y_2) = \left(\frac{1}{3}, \frac{1}{3}\right)\).

Step 5: Use Composite Area Formula for Center of Mass

The overall center of mass \((X, Y)\) of the region \(\mathcal{R}\) is found using the formula: \[ X = \frac{A_1 x_1 + A_2 x_2}{A_1 + A_2}, \quad Y = \frac{A_1 y_1 + A_2 y_2}{A_1 + A_2} \]Substitute known values: \( X = \frac{1(-0.5) + \frac{1}{2}(\frac{1}{3})}{1 + \frac{1}{2}} = \frac{-0.5 + \frac{1}{6}}{\frac{3}{2}} = -\frac{1}{3}\) \( Y = \frac{1(0.5) + \frac{1}{2}(\frac{1}{3})}{1 + \frac{1}{2}} = \frac{0.5 + \frac{1}{6}}{\frac{3}{2}} = \frac{1}{2}\) Thus, the center of mass is \((-\frac{1}{3}, \frac{1}{2})\).

Key Concepts

Composite Area Formula

When calculating the center of mass for a shape that is made up of multiple simple geometric forms, like rectangles and triangles, we use the composite area formula. It helps predict where the center of mass (centroid) of the entire shape lies by considering the areas and the centroids of its component parts.
For example, the formula to find the center of mass coordinates \(X, Y\) is given by:

• For X-axis: \[ X = \frac{\sum (A_i x_i)}{\sum A_i} \]
• For Y-axis: \[ Y = \frac{\sum (A_i y_i)}{\sum A_i} \]

Here, \(A_i\) represents the area of each component shape, and \(x_i\) and \(y_i\) are the respective x and y coordinates of their centroids.
This technique is crucial in simplifying the process of finding a shape's center of mass by effectively leveraging symmetry and uniformity of the shapes.

Uniform Density

Uniform density is a key consideration when finding the center of mass. It means that the mass is spread evenly throughout the shape. This concept simplifies the calculation, as you can assume that mass is proportionally distributed according to area.
In practice, this means:

• Each unit area contributes equally to the mass of the entire shape.
• You can calculate the center of mass without separately weighing each section.

In problems where uniform density is assumed, any calculations of mass-related elements, like the center of mass, rely solely on geometrical properties and measurements, not material differences or varying thickness.

Geometric Shapes

In the context of finding the center of mass, understanding basic geometric shapes like rectangles and triangles is vital. Each shape has unique properties and formulas to determine crucial points like centroids.
**Key Points About Shapes**

• **Rectangle**: Regular four-sided shape. Its centroid is located symmetrically at the intersection of its diagonals which is simply at the center of the rectangle.
• **Triangle**: Its centroid, often called the center of gravity, is found at the average of its vertices' coordinates or can be computed by consistent formulas.

Knowledge of these basic shapes helps in calculating the areas and knowing the position of their centroids which is crucial in the composite area formula.

Rectangle and Triangle Area Calculation

Calculating the area of basic shapes like rectangles and triangles is foundational in determining the center of mass for complex shapes.**Rectangle Area**:

• The area of a rectangle is straightforward—product of its length and width. For a rectangle spanning from \(x = a\) to \(x = b\) and within the \(y = c\) plane, its area will be \(\text{Area} = (b-a) \times c\).

**Triangle Area**:

• The area formula \( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \) helps ensure correct area calculations for triangles, which are often parts of larger regions.

Accurate computation of these areas allows for precise determinations of the center of mass when using the composite area formula.