Problem 33
Question
Let \(S_{1}\) and \(S_{2}\) be disjoint laminas in the \(x y\) -plane of mass \(m_{1}\) and \(m_{2}\) with centers of mass \(\left(\bar{x}_{1}, \bar{y}_{1}\right)\) and \(\left(\bar{x}_{2}, \bar{y}_{2}\right) .\) Show that the center of mass \((\bar{x}, \bar{y})\) of the combined lamina \(S_{1} \cup S_{2}\) satisfies $$ \bar{x}=\bar{x}_{1} \frac{m_{1}}{m_{1}+m_{2}}+\bar{x}_{2} \frac{m_{2}}{m_{1}+m_{2}} $$ with a similar formula for \(\bar{y}\). Conclude that in finding \((\bar{x}, \bar{y})\) the two laminas can be treated as if they were point masses at \(\left(\bar{x}_{1}, \bar{y}_{1}\right)\) and \(\left(\bar{x}_{2}, \bar{y}_{2}\right)\).
Step-by-Step Solution
Verified Answer
Combine lamina center of mass locations as if they are point masses, using weighted averages.
1Step 1: Calculate Total Mass
Sum the masses of both laminas to find the total mass of the system. Since the laminas are disjoint, the total mass is simply the sum of the individual masses: \[ m = m_1 + m_2 \]
2Step 2: Determine the Weighted X-Coordinate
To find the x-coordinate of the center of mass for the combined system, use the formula: \[ \bar{x} = \frac{m_1 \bar{x}_1 + m_2 \bar{x}_2}{m_1 + m_2} \]This formula takes into account the masses and x-coordinates of each lamina, resulting in a weighted average based on their respective masses.
3Step 3: Simplify the X-Coordinate Formula
Simplify the expression for \( \bar{x} \): \[ \bar{x} = \frac{m_1 \bar{x}_1}{m_1 + m_2} + \frac{m_2 \bar{x}_2}{m_1 + m_2} \]This shows that each x-coordinate is weighted by its respective mass divided by the total mass.
4Step 4: Determine the Weighted Y-Coordinate
Find the y-coordinate of the center of mass in a similar way. Use the formula: \[ \bar{y} = \frac{m_1 \bar{y}_1 + m_2 \bar{y}_2}{m_1 + m_2} \]Again, this is a weighted average of the y-coordinates based on the masses.
5Step 5: Simplify the Y-Coordinate Formula
Simplify the expression for \( \bar{y} \): \[ \bar{y} = \frac{m_1 \bar{y}_1}{m_1 + m_2} + \frac{m_2 \bar{y}_2}{m_1 + m_2} \]This confirms that the y-coordinate follows a similar weighted average formula as the x-coordinate.
6Step 6: Conclude with the Interpretation
The expressions for \( \bar{x} \) and \( \bar{y} \) show that the center of mass of the combined lamina behaves as if the individual laminas are point masses located at their respective centers of mass.
Key Concepts
laminasweighted averagepoint massesmass distribution
laminas
Laminas are flat, two-dimensional objects with mass distributed over their surface area. In problems involving laminas in the plane, such as the one above, two separate laminas are considered (denoted as \(S_1\) and \(S_2\)). Each lamina possesses its own mass and center of mass, much like a thin sheet across a tabletop.
The concept of a lamina is helpful because it simplifies real-world objects to a manageable, 2D representation suitable for analysis in plane geometry. For each lamina, its mass \(m\) is concentrated evenly across its surface despite its shape or size.
This allows us to consider each lamina's weight and how it affects the balance of the entire structure, which is particularly useful in calculations of combined centers of mass.
The concept of a lamina is helpful because it simplifies real-world objects to a manageable, 2D representation suitable for analysis in plane geometry. For each lamina, its mass \(m\) is concentrated evenly across its surface despite its shape or size.
This allows us to consider each lamina's weight and how it affects the balance of the entire structure, which is particularly useful in calculations of combined centers of mass.
weighted average
The weighted average is a crucial tool when calculating centers of mass, as it determines how much influence each component part exerts in finding a shared center. In this context, the x and y coordinates of the laminas' centers of mass are used along with their masses to calculate the overall center of mass.
The formula for the weighted average involves summing the products of each component's value (such as coordinates) and its weight (in this case, its mass), and then dividing by the total weight. For example, the formula for the x-coordinate of the center of mass is:
Using the weighted average ensures that the total effect of varying masses is accurately captured, aiding in precise balance calculations.
The formula for the weighted average involves summing the products of each component's value (such as coordinates) and its weight (in this case, its mass), and then dividing by the total weight. For example, the formula for the x-coordinate of the center of mass is:
- \[ \bar{x} = \frac{m_1 \bar{x}_1 + m_2 \bar{x}_2}{m_1 + m_2} \]
Using the weighted average ensures that the total effect of varying masses is accurately captured, aiding in precise balance calculations.
point masses
In this exercise, laminas can be effectively treated as point masses located at their respective centers of mass. This interpretation simplifies the calculations of centers of mass across multiple laminas.
A point mass is an idealization where the mass of an object is concentrated at a single point. This abstraction makes it easier to compute the center of mass for complex systems because the geometry of the object doesn’t need to be further considered.
By simplifying each lamina to a point mass, you still accurately calculate the combined center of mass, assuming all mass is concentrated at the center of its corresponding lamina. It translates multifaceted shapes or objects into manageable mathematical points, thereby streamlining the steps needed to solve problems involving multiple objects.
A point mass is an idealization where the mass of an object is concentrated at a single point. This abstraction makes it easier to compute the center of mass for complex systems because the geometry of the object doesn’t need to be further considered.
By simplifying each lamina to a point mass, you still accurately calculate the combined center of mass, assuming all mass is concentrated at the center of its corresponding lamina. It translates multifaceted shapes or objects into manageable mathematical points, thereby streamlining the steps needed to solve problems involving multiple objects.
mass distribution
Mass distribution refers to how mass is spread out over the surface or throughout the volume of an object. In the context of centers of mass involving laminas, mass distribution is significant because it influences where the center, or balance point, of the mass lies.
For laminas, which are considered uniform, mass distribution is even across their surfaces. This uniformity simplifies computational tasks as it conforms to the assumptions made when calculating centers of mass. The calculation assumes that the mass is symmetrically spread, affecting how the weighted average is performed.
Understanding mass distribution aids in comprehending how and why the centers of mass are calculated in a certain way, as it directly relates to how weight influences balance and positioning in physical systems.\( \)
For laminas, which are considered uniform, mass distribution is even across their surfaces. This uniformity simplifies computational tasks as it conforms to the assumptions made when calculating centers of mass. The calculation assumes that the mass is symmetrically spread, affecting how the weighted average is performed.
Understanding mass distribution aids in comprehending how and why the centers of mass are calculated in a certain way, as it directly relates to how weight influences balance and positioning in physical systems.\( \)
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