Question

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A tetrahedron is bounded by the coordinate planes and the plane \(x / a+y / a+z / a=1 .\) What are the coordinates of the center of mass?

Short answer

Answer: The coordinates of the center of mass are \(\left(\frac{1}{4}a,\frac{1}{4}a,\frac{1}{4}a\right)\).

Step-by-step solution

Geometry of the Tetrahedron

The tetrahedron is bounded by the coordinate planes \((x = 0, y = 0, z = 0),\) and the plane \(x / a+y / a+z / a=1\). The vertices of the tetrahedron are given by the intersection of these planes. The vertices are located at \((0, 0, 0)\), \((a, 0, 0)\), \((0, a, 0)\), and \((0, 0, a)\).

Choose a convenient coordinate system

The coordinate system provided (Cartesian) is suitable for this problem as the bounding planes are given in Cartesian coordinates.

Calculate Center of Mass Coordinates

Assuming constant density, the center of mass coordinates are the averages of the coordinates of the vertices. Let's separately compute the x, y, and z coordinates of the center of mass: - X coordinate: \(\frac{x_1 +x_2 +x_3 +x_4}{4} = \frac{0+a+0+0}{4} = \frac{a}{4}\) - Y coordinate: \(\frac{y_1 +y_2 +y_3 +y_4}{4} = \frac{0+0+a+0}{4} = \frac{a}{4}\) - Z coordinate: \(\frac{z_1 +z_2 +z_3 +z_4}{4} = \frac{0+0+0+a}{4} = \frac{a}{4}\) Thus, the coordinates of the center of mass are \(\left(\frac{1}{4}a,\frac{1}{4}a,\frac{1}{4}a\right)\).

Key concepts

Tetrahedron

A tetrahedron is a polyhedron with four triangular faces, six edges, and four vertices. It is sometimes referred to as a "triangular pyramid". This geometric shape is one of the simplest forms of a three-dimensional object. A tetrahedron is special because it is a form found commonly in nature and can be seen in complex structures like molecular bonds.

In the context of the problem, the tetrahedron is bounded by the three coordinate planes, which are the xy-plane, yz-plane, and xz-plane. These correspond to the conditions where one coordinate is zero, like the planes where x, y, or z equals zero. Additionally, it is bounded by the plane equation \(\frac{x}{a} + \frac{y}{a} + \frac{z}{a} = 1\). This equation defines a plane in three-dimensional space. The tetrahedron formed under these parameters has its vertices at points \( (0, 0, 0)\), \( (a, 0, 0)\), \( (0, a, 0)\), and \( (0, 0, a)\).

Understanding the geometry of a tetrahedron helps in visualizing the problem and figuring out physical properties like volume or in our case, finding the center of mass by averaging the vertices.

Coordinate System

A coordinate system is a system that uses numbers to uniquely determine the position of a point. It is essentially a framework used to map geometric figures into numbers, which simplifies calculations and visualization.

In our problem, we use a **Cartesian coordinate system**, which is one of the most common types of coordinate systems used in mathematics and physics. It involves three axes: x, y, and z, that are perpendicular to each other and meet at a point (called the origin). These axes divide the space into a three-dimensional grid which makes it easier to determine the positions and relationships between points.

Choosing the right coordinate system is crucial when solving geometry problems as certain shapes align better with particular systems. The Cartesian system was chosen here because the problem already defines the bounding surfaces using Cartesian equations, making it easier to apply mathematical operations on them. The clear definition of the vertices in Cartesian coordinates also paves the way for the straightforward calculation of the center of mass based on those points.

Cartesian Coordinates

Cartesian coordinates are used to determine the position of a point in 2D, 3D, or even higher dimensions based on numeric values for the axes. They are named after the mathematician René Descartes and have been foundational in the field of geometry and algebra.

In this exercise, Cartesian coordinates help us express the location of the vertices of the tetrahedron, which are \( (0, 0, 0)\), \( (a, 0, 0)\), \( (0, a, 0)\), and \( (0, 0, a)\). Each vertex can be described as a combination of the coordinates along the x, y, and z axes. These coordinates simplify the calculation of the center of mass because they permit easy averaging.

Calculating the center of mass in Cartesian coordinates involves taking the average of each of the coordinate axes separately. For this tetrahedron, the x, y, and z coordinates of the center of mass were derived by averaging each set of respective vertex coordinates. Since each coordinate axis contributes equally due to symmetry and constant density, the calculated center of mass coordinates reflect the balance of the mass around each axis.