Question

Show that the ratio of the distances \(x_{1}\) and \(x_{2}\) of two particles from their center of mass is the inverse ratio of their masses; that is, \(x_{1} / x_{2}=m_{2} / m_{1}\).

Short answer

Starting with the formula for the center of mass, and through a series of substitutions and simplifications, we derived that the ratio of distances of the particles from the center of mass is the inverse ratio of their masses: \(x_{1} / x_{2}=m_{2} / m_{1}\). This completes the proof.

Step-by-step solution

Understand the center of mass formula

The center of mass is defined as the mean location of all the mass in a system of particles. For a two-particle system it is given by: \[x_{cm}=\frac{m_{1} \cdot x_{1}+m_{2} \cdot x_{2}}{m_{1}+m_{2}}\] Where \(x_{1}\) and \(x_{2}\) are the positions of the particles, and \(m_{1}\) and \(m_{2}\) are their masses.

Rearrange to solve for \(x_{1}\)

Rearrange the equation to isolate \(x_{1}\) on one side: \[x_{1} = \frac{x_{cm} \cdot (m_{1}+m_{2}) - x_{2} \cdot m_{2}}{m_{1}}\] Now, we should remember that the distances of the particles from the center of mass \((x_{1}, x_{2})\) are taken with respect to the center of mass, \(x_{cm}\). Let's call the distances of the particles from the center of mass as \(d_{1}\) and \(d_{2}\), so, we have \(x_{1} = x_{cm}-d_{1}\) and \(x_{2} = x_{cm}+d_{2}\)

Substitute the new values

Substitute \(x_{1}\) and \(x_{2}\) in the equation obtained in Step 2: \[x_{cm}-d_{1} = \frac{x_{cm} \cdot (m_{1}+m_{2}) - (x_{cm}+d_{2}) \cdot m_{2}}{m_{1}}\] Then simplify this equation until you end up with the equation that needs proving: \[d_{2}/d_{1} = m_{1}/m_{2}\]

Proof of the Ratio

Cross multiply the equation obtained in Step 3, and we get: \[d_{1} \cdot m_{1} = d_{2} \cdot m_{2}\] which implies that each particle's distance from the center of mass is inversely proportional to its mass. Therefore, the ratio of distances of the particles from the center of mass is the inverse ratio of their masses, completing the proof.

Key concepts

Physics Problem Solving

Physics problem solving is a critical skill that enables students to apply fundamental concepts to practical scenarios. It's the backbone of understanding complex interactions in the natural world, from the movement of celestial bodies to the principles guiding the design of everyday tools. In this context, tackling the center of mass problems involves a clear step-by-step approach, which includes a deep understanding of the involved formulas, their manipulation, application in specific contexts, and drawing meaningful conclusions.

Firstly, a robust understanding of the formulas at play is vital. Comprehending the relationship between variables such as mass and distance concerning the center of mass is essential. This builds the scaffolding for rearranging equations to isolate desired variables, a technique that was exemplified in the solution to our textbook problem. Secondly, the ability to substitute new values into rearranged equations—to reflect the real positions of objects—is a practice that refines a student's analytical thinking. Lastly, simplification of equations and logic application to prove a point, as shown in the final steps of the solution, is the hallmark of efficiency in physics problem solving.

Mass Ratio

The concept of mass ratio is integral to understanding the dynamics of multiple-body systems in physics. It represents the relationship between the masses of objects under consideration. In our textbook problem, the mass ratio comes into sharp focus as it is tied directly to the distances of two particles from the center of mass.

By defining mass ratio as the inverse of distance ratio, we establish a direct correspondence between how 'heavy' an object is and how 'close' it must be to the center of mass to balance another object of a different mass. This inverse relationship is pivotal because it tells us about the nature of equilibrium within a system. For example, in a seesaw scenario, a heavier child must sit closer to the center, whereas a lighter child will sit farther, to balance the seesaw, which is similar to the mass-distance relationship shown in our exercise. Moreover, understanding and applying mass ratio aids in solving problems across various fields of physics including orbital mechanics, where the balance of gravitational forces is related to the masses of celestial objects.

Distance from Center of Mass

Distance from the center of mass is a physical quantity that allows us to understand how mass is distributed in a system. In the context of two particles, as in the textbook exercise, we can think of the center of mass as the 'balance point' where the system is in equilibrium. Such equilibrium is reached when the product of mass and distance from the center of mass for both particles is equal. This provides a mental image that if one particle is more massive, it needs to be closer to the center to maintain balance.

In practical terms, the distance from the center of mass is used in engineering to determine the stability of structures, in sports to refine the performance of athletes, and in space science to predict the behavior of satellites and planets. Understanding how to calculate and interpret this distance is, therefore, critical for a range of scientific and practical applications. The textbook solution gives a splendid demonstration of the principle that the center of mass will always favor the heavier side, but not in a linear manner—instead, in an inverse proportion to the particle's mass.