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? $\\{\mathrm{A}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]$ $\\{\mathrm{B}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]$ $\\{\mathrm{C}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{2}\right)\right]$ $\\{\mathrm{D}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}\right)\right]$

Step by step solution

01

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.

02

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: \(r_1' = -d\)

03

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: \(\frac{M_1(0) + M_2r_2}{M_1 + M_2} = \frac{M_1(-d) + M_2r_2'}{M_1 + M_2}\)

04

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): \(\frac{M_1d}{M_2}\)

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