Question

The centre of mass of a systems of two particles is (A) on the line joining them and midway between them (B) on the line joining them at a point whose distance from each particle is proportional to the square of the mass of that particle. (C) on the line joining them at a point whose distance from each particle inversely proportional to the mass of that particle. (D) On the line joining them at a point whose distance from each particle is proportional to the mass of that particle.

Short answer

The correct answer is (D). The center of mass is on the line joining the two particles, and the distance from each particle is proportional to the mass of that particle.

Step-by-step solution

Analyzing Statement A

Statement A says that the center of mass is on the line joining the two particles and midway between them. This statement would be true if the masses of the two particles were equal, i.e., \(m_1 = m_2\). In this case, the center of mass formula becomes: \[R_{cm} = \frac{m_1 r_1 + m_1 r_2}{m_1 + m_1} = \frac{2m_1 (r_1 + r_2)}{2m_1} = \frac{r_1 + r_2}{2}\] However, since the masses are not specified to be equal, statement A is not necessarily true.

Analyzing Statement B

Statement B says that the center of mass is on the line joining the two particles, and the distance from each particle is proportional to the square of the mass of that particle. The center of mass formula does not include squares of the masses, so statement B is incorrect.

Analyzing Statement C

Statement C says that the center of mass is on the line joining the two particles, and the distance from each particle is inversely proportional to the mass of that particle. Let's designate the distance from the center of mass to particles 1 and 2 as \(d_1\) and \(d_2\), respectively. According to statement C, \(d_1 \propto \frac{1}{m_1}\) and \(d_2 \propto \frac{1}{m_2}\). Now, let's write \(d_1\) and \(d_2\) in terms of \(r_1\) and \(r_2\): \[d_1 = r_2 - R_{cm}\] and \[d_2 = R_{cm} - r_1\] Replacing the center of mass formula in \(d_1\) and \(d_2\): \[d_1 = r_2 - \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}\] \[d_2 = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2} - r_1\] From the center of mass formula, we cannot conclude that the distances from the center of mass to the particles are inversely proportional to their masses, so statement C is incorrect.

Analyzing Statement D

Statement D says that the center of mass is on the line joining the two particles, and the distance from each particle is proportional to the mass of that particle. Returning to the distances \(d_1\) and \(d_2\) from statement C: \[d_1 \propto m_1\] and \[d_2 \propto m_2\] Multiplying both sides of the first equation yields: \[d_1 = k_1 m_1\] Repeating this process for the second equation: \[d_2 = k_2 m_2\] We have: \[\frac{d_1}{d_2} = \frac{k_1 m_1}{k_2 m_2}\] Since both distances are from the same reference point (i.e., the center of mass), we have: \[\frac{R_{cm} - r_1}{r_2 - R_{cm}} = \frac{m_1}{m_2}\] Comparing this equation with the center of mass formula, we see that statement D is correct.