Question

Consider a two-particle system with the particles having masses \(\mathrm{M}_{1}\), and \(\mathrm{M}_{2}\). If the first particle is pushed towards the centre of mass through a distance \(d\), by what distance should the second particle be moved so as to keep the centre of mass at the same position? \(\\{\mathrm{A}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{B}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{C}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{D}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}\right)\right]\)

Short answer

The short answer is: The second particle should be moved by a distance of \(\frac{M_1d}{M_2}\) to keep the center of mass at the same position.

Step-by-step solution

Identify the initial center of mass

We will begin by calculating the initial center of mass for the two-particle system. The center of mass can be calculated using the following formula: \(r_{CM} = \frac{M_1r_1 + M_2r_2}{M_1 + M_2}\) Here, r1 and r2 are the positions of the first and second particles, respectively.

Calculate the new position of the first particle

The problem states that the first particle is being pushed towards the center of mass by a distance d. Let's assume that the first particle is initially at the origin, i.e., r1 = 0. The new position of the first particle will now be: \(r_1' = -d\)

Find the condition for the center of mass to be constant

The center of mass must remain constant, i.e., the new center of mass rCM' should be equal to rCM. From Step 1, we can write the new center of mass as: \(r_{CM}' = \frac{M_1r_1' + M_2r_2'}{M_1 + M_2}\) As the center of mass remains constant, we can assert: \(r_{CM} = r_{CM}'\) Plugging in the values from Step 1 and Step 2, and replacing r1 with 0, we get: \(\frac{M_1(0) + M_2r_2}{M_1 + M_2} = \frac{M_1(-d) + M_2r_2'}{M_1 + M_2}\)

Solve for the new position of the second particle

Now, we solve the equation obtained in Step 3 for r2': \(M_1(0) + M_2r_2 = M_1(-d) + M_2r_2'\) \(M_2r_2 = -M_1d + M_2r_2'\) \(r_2' = \frac{M_1d}{M_2} + r_2\) Since we want to find the distance the second particle must move and not its new position, we need to calculate the difference between r2' and r2: Distance required = \(r_2' - r_2 = \frac{M_1d}{M_2}\) The correct answer is given by the option (C): \(\frac{M_1d}{M_2}\)

Key concepts

two-particle system

In physics, a two-particle system refers to a system that consists of two interacting particles. These particles can have different masses and are often studied to understand how they influence each other's movements due to forces acting upon or between them.
In this exercise, we examine a scenario where two particles have masses labeled as \(M_1\) and \(M_2\). These particles can be envisioned on a line, and their movements can affect the system's overall center of mass. Understanding such systems is fundamental in physics as it roots back to basic principles of motion and equilibrium calculations. Here, the key point is to analyze the effect of displacing one particle on the system and how to counterbalance it with the displacement of the other particle to maintain the system's properties.

conservation of center of mass

The principle of conservation of center of mass is crucial in many physics problems. This principle states that the center of mass of a system remains constant if no external forces act on it.
In our scenario, the center of mass is derived using the formula \(r_{CM} = \frac{M_1r_1 + M_2r_2}{M_1 + M_2}\).
This gives you a weighted average position based on the mass and position of the particles. When moving one of the particles, the center of mass only remains unchanged if the product of mass and displacement is counterbalanced by the movement of the other particle.

• This property ensures system equilibrium.
• Understanding how to maintain a constant center can aid in simulations of complex, many-body problems.

This exercise demonstrates conservational dynamics even in relatively simple systems, which becomes a stepping stone for more advanced problems involving multi-particle interactions.

mass displacement

Mass displacement is a critical concept when discussing movements relative to the center of mass. Displacement refers to how far and in what direction a mass moves from its original position.
In this exercise, we start by moving the first particle by a distance \(d\) towards the center of mass. This shift creates an imbalance in the system. To restore balance and conserve the center of mass, the second particle must also be moved.

• Using the solution, we learned that the second particle needs to move a distance \(\frac{M_1d}{M_2}\).
• This calculation depends solely on the ratio of the masses and the initial displacement \(d\).

Therefore, the task becomes finding the appropriate adjustment that accounts for both mass and distance, a core aspect of solving these types of physics problems.

physics problem solving

Physics problem solving often involves breaking down complex systems into manageable parts and using known principles to find solutions.
For our two-particle system, we start by recognizing and defining the core principles, such as conservation of center of mass, before applying formulas to seek a solution.
Successful problem-solving requires:

• A clear grasp of fundamental concepts.
• Systematic approach - often involving step-by-step calculation.
• Applying relevant equations perfectly suited to the issue at hand.

In this example, isolating the displacement calculations from mass considerations and ensuring their symmetric balance guided us to the solution. Through this exercise, one builds skills applicable to broader contexts, as problem-solving in physics frequently involves such logical and methodological patterns.