Question

Find the \(x\) - and \(y\) -coordinates of the center of mass of the flat triangular plate of height \(H=17.3 \mathrm{~cm}\) and base \(B=10.0 \mathrm{~cm}\) shown in the figure.

Short answer

Answer: The coordinates of the center of mass of the flat triangular plate are approximately (0, 5.8).

Step-by-step solution

Label the triangle vertices

We label the vertices of the triangle as \(A\), \(B\), and \(C\) with \(A\) at the top vertex of the triangle and \(B\) and \(C\) at the base. Also, let the midpoint of the side \(BC\) be \(D\). So, \(\overline{AD}\) is a median of the triangle.

Calculate the midpoint D

The coordinates of the vertices are: \(A(0, 17.3),\) \(B(-5, 0),\) and \(C(5, 0)\) since the base is \(10.0 \mathrm{~cm}\) and the height is \(17.3 \mathrm{~cm}\). To calculate the midpoint, we apply the midpoint formula for \(x\) and \(y\) coordinates: \(x_{mid} = \frac{x_1 + x_2}{2}\) and \(y_{mid} = \frac{y_1 + y_2}{2}\) By substituting the coordinates of points \(B\) and \(C\), coordinates of midpoint \(D\) are: \(x_{mid}= \frac{-5 + 5}{2} = 0\) \(y_{mid}= \frac{0 + 0}{2} = 0\) So, the midpoint \(D\) has the coordinates \((0,0)\).

Calculate the coordinates of the center of mass

Now, we know that the center of mass of a triangle lies on the median and is \(2/3\) of the distance from the vertex to the midpoint. In our case, this is the distance between \(A\) and \(D\). So, the coordinates of the center of mass \((x_{cm}, y_{cm})\) can be calculated using the following formulas: \(x_{cm}=x_A + \frac{2}{3}(x_D - x_A)=0+\frac{2}{3}(0-0)=0\) \(y_{cm}=y_A + \frac{2}{3}(y_D - y_A)=17.3+\frac{2}{3}(0-17.3)=17.3-\frac{2}{3}\times 17.3\approx 5.8\) Hence, the coordinates of the center of mass of this flat triangular plate are \((x_{cm}, y_{cm})=(0, 5.8)\).

Key concepts

Understanding Physics in Finding the Center of Mass

Physics is a vast field that deals with the study of matter, energy and the relationships between them. A fundamental concept in physics is the center of mass, which simplifies the analysis of motion for complex systems. When it comes to objects like a flat triangular plate, finding the center of mass allows us to predict how the object will behave under various forces.

Think of the center of mass as a point where we could balance the triangle on the tip of a pencil without it tipping over. The entire mass of the triangle seems to be concentrated at this point, and it is the point that moves in space as if all external forces were applied there. Understanding this physics concept is essential when solving problems involving stability, rotation, and gravity's influence on objects of various shapes and sizes.

Mechanics and the Center of Mass

Mechanics is the branch of physics that deals with the behavior of physical bodies when subjected to forces or displacements. It's split into statics, which studies bodies at rest or in equilibrium, and dynamics, which studies bodies in motion. The center of mass falls under the umbrella of mechanics because it's a critical concept that affects how bodies balance, move, and interact.

In the context of the triangular plate problem, mechanics helps us understand why the center of mass is located at a certain point and how this affects the plate's stability. For instance, a triangular plate hanging from its center of mass will remain at rest, illustrating balance as described in statics, while if it's thrown into the air, the plate will rotate around its center of mass, exemplifying concepts from dynamics.

Coordinate System in the Context of Center of Mass

A coordinate system is a method used in geometry and physics to uniquely determine the position of a point or a geometric element in a multidimensional space. In our case, we are using a two-dimensional coordinate system to determine the position of the center of mass of the triangular plate.

The Cartesian coordinate system is the most common and consists of two perpendicular axes: horizontal (x-axis) and vertical (y-axis). Each point in this system is represented by an ordered pair of numbers, the coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. With respect to our triangular plate, the center of mass is described by its location along these x and y axes, providing a clear and precise way to define its position in space. This is an invaluable tool for solving problems in physics and various branches of engineering.

Median of a Triangle and the Center of Mass

The median of a triangle is a line segment that joins a vertex of the triangle to the midpoint of the opposite side. Every triangle has three medians, and an interesting property is that they all intersect at one point, which is the center of mass (also known as centroid) of the triangle. This point is unique to each triangle and has many useful properties.

For the purpose of our problem, we used the median to locate the center of mass of a triangular plate. We calculated the midpoint of the base and drew a line from the vertex to this midpoint - that's our median. By understanding that every median divides the triangle into two smaller triangles of equal area, we further understand why the intersection of medians - the centroid - truly represents the average location of the mass distribution. Knowing that the centroid is located 2/3 of the way from each vertex to the opposite side's midpoint provides us with a precise mathematical method to find the center of mass, which is exactly what we did for our triangle.