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Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions (a) John has a larger mass than Barbara has. He is standing on the \(x\) axis at \(x_{\mathrm{J}}=+9.0 \mathrm{~m},\) while she is standing on the \(x\) axis at \(x_{\mathrm{B}}=+2.0 \mathrm{~m} .\) Is their centerof-mass point closer to the 9.0 -m point or the 2.0 -m point? (b) They switch positions. Is their center- of-mass point now closer to the 9.0 -m point or the 2.0 -m point? (c) In which direction, toward or away from the origin, does their center of mass move as a result of the switch? Problem John's mass is \(86 \mathrm{~kg}\), and Barbara's is \(55 \mathrm{~kg}\). How far and in which direction does their center of mass move as a result of the switch? Verify that your answer is consistent with your answers to the Concept Questions.

Step by step solution

01

Explain Center of Mass Concept

The center of mass is a point that represents the average position of the total mass of a system. For two objects along an axis, the center of mass can be found using the formula: \( x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \). Where \( m_1 \) and \( m_2 \) are the masses, and \( x_1 \) and \( x_2 \) are their positions.

02

Evaluate Concept Question (a)

Since John has more mass than Barbara and is at \(+9.0 \mathrm{~m}\), the center of mass will be closer to John's position. This is because the center of mass is weighted more towards the object with the larger mass.

03

Evaluate Concept Question (b)

After switching positions, John is now at \(+2.0 \mathrm{~m}\) and Barbara at \(+9.0 \mathrm{~m}\). The center of mass will now shift closer to John's new position at \(+2.0 \mathrm{~m}\) since he still has a larger mass than Barbara.

04

Evaluate Concept Question (c)

Given that John moves from \(+9.0 \mathrm{~m}\) to \(+2.0 \mathrm{~m}\) and Barbara from \(+2.0 \mathrm{~m}\) to \(+9.0 \mathrm{~m}\), the center of mass will move away from the origin since it primarily shifts towards John's position, which is now closer to the origin.

05

Calculate Initial Center of Mass

Using the center of mass formula: \( x_{\text{cm}} = \frac{86 \times 9.0 + 55 \times 2.0}{86 + 55} = \frac{774 + 110}{141} = \frac{884}{141} \approx 6.27 \mathrm{~m} \).

06

Calculate New Center of Mass After Switch

After switching: \( x_{\text{cm}} = \frac{86 \times 2.0 + 55 \times 9.0}{86 + 55} = \frac{172 + 495}{141} = \frac{667}{141} \approx 4.73 \mathrm{~m} \).

07

Determine Center of Mass Movement

The center of mass moves from approximately \(6.27 \mathrm{~m}\) to \(4.73 \mathrm{~m}\). This confirms the movement is towards the origin, consistent with the predictions from the concept questions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Distribution

In physics, understanding the distribution of mass within a system is essential for solving problems, particularly those involving the center of mass. The center of mass is a point that represents where we can consider the entire mass of a system to be concentrated. For two or more objects, the mass distribution affects this point significantly. The heavier an object in the system, the more influence it has on the position of the center of mass. As with John and Barbara, since John has more mass, his position plays a greater role in determining where the center of mass lies along the x-axis. Thus, mass distribution gives us insight into why the center of mass would initially be closer to the heavier object's position, before any changes or switches in position.

Physics Problem Solving

Physics problem-solving often begins with simplifying complex scenarios into manageable calculations and understanding their underlying concepts. Tackling problems like those involving the center of mass requires a step-by-step approach.

When John and Barbara change positions, the solution is not just about crunching numbers. It requires an understanding of basic physics principles. One must first recognize that the center of mass depends on both the mass's magnitude and position. Then, by applying the center-of-mass formula, one can determine how their switch affects the overall balance of the system. Working through such problems helps students reinforce their understanding of fundamental physics concepts by always grounding their calculations in real-world analogies and scenarios.

Concept Questions

Concept questions, like those before solving John's and Barbara's problem, are crucial in highlighting the theoretical understanding needed before diving into numerical solutions. They engage the student's ability to think about the problem conceptually, without immediate reliance on mathematics.

In the case of John and Barbara, concept questions help illustrate how mass and position affect the center of mass location. They prompt students to consider qualitative outcomes, such as whether the center of mass is closer to one person's position and in which direction it moves when they switch places. These questions aid students in forming a mental model of the problem, making it easier to approach quantitative problem-solving with confidence and clarity.

Mathematical Analysis

Mathematical analysis in physics provides the tools necessary to quantify abstract concepts into concrete values. By using the center of mass formula, students can determine the exact new position of the center of mass when mass distributions change. In the given exercise, mathematical analysis translates the positions and mass of John and Barbara into a precise location of the center of mass.

Calculations show the initial center of mass at approximately 6.27 m and, after switching positions, at about 4.73 m. This result aligns with our understanding from the concept questions, confirming the usefulness of mathematical analysis as a technique to check logical assumptions within a physics problem. It helps bridge the gap between theory and practice, providing a model for predictive accuracy.