Question

A thin rectangular plate of uniform area density \(\sigma_{1}=1.05 \mathrm{~kg} / \mathrm{m}^{2}\) has a length \(a=0.600 \mathrm{~m}\) and a width \(b=0.250 \mathrm{~m} .\) The lower left corner is placed at the origin, \((x, y)=(0,0) .\) A circular hole of radius \(r=0.048 \mathrm{~m}\) with center at \((x, y)=(0.068 \mathrm{~m}, 0.068 \mathrm{~m})\) is cut in the plate. The hole is plugged with a disk of the same radius that is composed of another material of uniform area density \(\sigma_{2}=5.32 \mathrm{~kg} / \mathrm{m}^{2}\) What is the distance from the origin of the resulting plate's center of mass?

Short answer

Answer: The distance from the origin of the resulting plate's center of mass is approximately 0.127 meters.

Step-by-step solution

Calculate the center of mass of the original plate

Since the plate is rectangular and has uniform density, its center of mass lies at the center of the rectangle. Therefore, the center of mass coordinates can be calculated as follows: $$(x_{1}, y_{1}) = (\frac{a}{2}, \frac{b}{2}) = (\frac{0.600}{2}, \frac{0.250}{2}) = (0.300, 0.125)\mathrm{~m}$$

Calculate the center of mass of the removed circular hole

As given, the circular hole has the center at \((0.068, 0.068)\), so the center of mass of this hole is: $$(x_{2}, y_{2}) = (0.068, 0.068) \mathrm{~m}$$

Calculate the center of mass of the inserted circular disk

Since the inserted disk has the same center and the same size as the removed hole, its center of mass is also at the same location: $$(x_{3}, y_{3}) = (0.068, 0.068) \mathrm{~m}$$

Calculate the total mass of each part

We need the mass of each part to find the center of mass of the resulting plate. For the original plate: $$m_1 = \sigma_1ab = (1.05 \mathrm{~kg/m^2})(0.600 \mathrm{~m})(0.250 \mathrm{~m}) = 0.1575 \mathrm{~kg}$$ For the removed circular hole: $$m_2 = \sigma_1\pi r^2 = (1.05 \mathrm{~kg/m^2})(\pi)(0.048 \mathrm{~m})^2 \approx 0.007558 \mathrm{~kg}$$ For the inserted disk: $$m_3 = \sigma_2\pi r^2= (5.32 \mathrm{~kg/m^2})(\pi)(0.048 \mathrm{~m})^2 \approx 0.384 \mathrm{~kg}$$

Calculate the center of mass of the resulting plate

Now, we have all the necessary information to find the location of the center of mass of the resulting plate, which is given by: $$(x_{\mathrm{CM}}, y_{\mathrm{CM}}) = \frac{(m_{1}x_{1} - m_{2}x_{2} + m_{3}x_{3}, m_{1}y_{1} - m_{2}y_{2} + m_{3}y_{3})}{m_1 - m_2 + m_3}$$ Substituting the values, we get: $$(x_{\mathrm{CM}}, y_{\mathrm{CM}}) \approx (\frac{(0.1575)(0.300) - (0.007558)(0.068) + (0.384)(0.068)}{0.1575 - 0.007558 + 0.384}, \frac{(0.1575)(0.125) - (0.007558)(0.068) + (0.384)(0.068)}{0.1575 - 0.007558 + 0.384}) \approx (0.0861, 0.0929) \mathrm{~m}$$

Calculate the distance from the origin to the center of mass

Finally, to find the distance from the origin to the center of mass, we use the distance formula: $$d = \sqrt{(x_{\mathrm{CM}}-0)^2 + (y_{\mathrm{CM}}-0)^2} \approx \sqrt{(0.0861)^2 + (0.0929)^2} \approx 0.127 \mathrm{~m}$$ The distance from the origin of the resulting plate's center of mass is approximately 0.127 meters.

Key concepts

Uniform Area Density

Understanding the concept of uniform area density is crucial when studying the center of mass in objects with continuous mass distribution. Area density, denoted by the symbol \( \sigma \), indicates how mass is spread over a two-dimensional (2D) area. If this distribution of mass is consistent across the entire object, we say that the object has a uniform area density. In simpler terms, it means each unit of area of the object has the same amount of mass.
For our rectangular plate example, the uniform area density \( \sigma_1 \) of 1.05 \(/\mathrm{kg/m}^2\) means that every square meter of the plate has a consistent mass of 1.05 kilograms. This uniformity simplifies the process of locating the center of mass because we can assume the mass is evenly distributed across the entire object, leading to the center of mass residing at the geometric center of the shape. If the density weren't uniform, finding the center of mass would be more complex and would require integrating the mass distribution across the object.

Mass Distribution

The mass distribution of an object describes how mass is spread throughout the object, which can affect the object's balance, stability, and motion. The distribution can be uniform or non-uniform; in objects with uniform mass distribution, the mass is evenly spread out, while non-uniform distribution means the mass is concentrated in certain areas more than others.
In our exercise, we encounter two mass distributions: the rectangular plate with uniform area density \( \sigma_1 \) and the circular disk with a different uniform area density \( \sigma_2 \). It's important to note how altering the mass distribution by substituting a portion of the rectangle with a higher-density disk changes the overall center of mass. This change demonstrates how even entities of equal shape but differing density can significantly influence the mass distribution and consequently the balance of the system. Whenever there is a substitution or removal of a part, we must adjust our calculations to reflect the modified mass distribution in order to find the new center of mass.

Coordinate System

Utilization of a coordinate system is essential in physics and engineering when locating positions in space. In this scenario, locating the center of mass of complex shapes is made manageable through the Cartesian coordinate system, which uses two perpendicular axes (usually labeled x and y) to define any point in a plane by its distance from these axes.
This system is immensely helpful for visualizing and solving the problem step by step. For instance, we defined the bottom left corner of the rectangular plate as the origin, \( (x, y) = (0,0) \), which serves as our reference point. From there, we could establish the coordinates of the center of mass for the original plate, the hole, and the disk. These coordinates were then used in accordance with the mass of the respective parts to compute the combined center of mass of the transformed plate. Thus, the Cartesian coordinate system is a powerful tool that provides a methodical approach to solve for the center of mass in compound objects.