Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

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

Answer

The center of mass \((\bar{x}, \bar{y})\) of the combined lamina can be treated as the weighted average of the individual centers of mass.

Step 1: Understand the Problem

We need to find the combined center of mass \((\bar{x}, \bar{y})\) of two disjoint laminas \(S_1\) and \(S_2\), which have individual masses and centers of mass. The key is to use the weighted average formula to derive the expression for the combined center of mass.

Step 2: Total Mass Calculation

Calculate the total mass of the system which is the sum of the masses of the two laminas: \[m = m_{1} + m_{2}\]

Step 3: Center of Mass Formula for x-coordinate

The x-coordinate of the center of mass of a system is a weighted average of the x-coordinates of the individual centers of mass. The formula is given by:\[\bar{x} = \frac{m_{1}\bar{x}_{1} + m_{2}\bar{x}_{2}}{m_{1} + m_{2}}\]

Step 4: Re-arrange to Desired Formula

Re-arrange the center of mass formula to match the given expression:\[\bar{x} = \bar{x}_{1}\frac{m_{1}}{m_{1} + m_{2}} + \bar{x}_{2}\frac{m_{2}}{m_{1} + m_{2}}\]

Step 5: Repeat for y-coordinate

Similarly, for the y-coordinate, use the formula:\[\bar{y} = \frac{m_{1}\bar{y}_{1} + m_{2}\bar{y}_{2}}{m_{1} + m_{2}} = \bar{y}_{1}\frac{m_{1}}{m_{1} + m_{2}} + \bar{y}_{2}\frac{m_{2}}{m_{1} + m_{2}}\]

Step 6: Conclusion

The formulation for \(\bar{x}\) and \(\bar{y}\) shows that both coordinates are a weighted average of the individual centers of mass of \(S_1\) and \(S_2\). This means you can treat the laminas as point masses located at their respective centers of mass.

Key Concepts

Laminas

A lamina is a two-dimensional object with constant thickness, like a sheet of paper or a thin metal plate. In mathematical problems, laminas are often used to simplify complex situations into planar problems, allowing us to focus on their mass distribution on a plane. The key feature of a lamina is that it has mass spread across its area. It is crucial to understand the concept of laminas when calculating the center of mass because the total mass and how it is distributed across the plane influence the center of mass calculation. The exercises involving laminas help us understand how to apply the center of mass concepts to flat surfaces.

Weighted Average

The concept of a weighted average is essential when dealing with the center of mass in systems like laminas. A weighted average takes into account the relative importance (or 'weight') of each value in a data set. The formula for the weighted average of two points looks like this:

• First, you find the sum of all weights (in our case, the masses: \(m_1 + m_2\)).
• Then, you multiply each point by its corresponding weight (\(m_1\bar{x}_1\) for the first lamina and \(m_2\bar{x}_2\) for the second).
• The weighted average is the sum of these products divided by the total mass.

Utilizing a weighted average allows us to calculate the precise center of mass of combined laminas, taking into account the mass and position of each lamina.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This branch of geometry allows us to understand geometric figures using algebraic equations.
When we calculate the center of mass of laminas, coordinate geometry helps us express the location of these centers in a logical, easy-to-use way. We use the coordinates \((\bar{x}_1, \bar{y}_1)\) and \((\bar{x}_2, \bar{y}_2)\), which represent the centers of mass of each lamina.
By using these coordinates in the center of mass formulas, we can derive precise expressions for the combined centers of mass, helping us to visualize and solve problems efficiently.
Coordinate geometry bridges the gap between abstract mathematical concepts and practical, visual applications, such as finding the optimal point where a lamina could balance based on its geometry and mass distribution.

Point Mass

A point mass is an idealized object of mass concentrated at a single point in space. In theoretical physics and mathematics, the concept of a point mass simplifies the study of systems by treating objects like the laminas in our exercise as if all their mass is located at one central point.

• When we calculate the center of mass of a system of laminas, we can treat each lamina as if it were a point mass located at its center of mass because these formulas ensure the accuracy of real-life predictions.
• This simplifies calculations, allowing us to apply formulas such as the weighted average more easily.

The assumption that a lamina can be reduced to a point mass makes it more straightforward to analyze its motion and balance, especially when dealing with systems composed of multiple such objects.
This conceptual simplification streamlines problem-solving and enhances our understanding of the dynamics of distributed mass systems.