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.

Key concepts

Two-Particle System

In physics, a **two-particle system** is quite straightforward yet reveals foundational insights about how several entities behave collectively. When we speak of a two-particle system, it means we're analyzing the attributes of two individual particles and how they interact as a whole. One critical concept here is the **center of mass**, which is the point in space where all the mass of the system seems to concentrate if you were to balance it.

Imagine two particles, each with their own mass and position in space. Instead of looking at all the complexity of their interactions, physicists simplify complex problems by focusing on the center of mass, which behaves as though all the mass were concentrated at this point for linear motion analysis. This simplification is vital when determining how forces will affect the system or how the system might move within a given reference frame. The center of mass for a two-particle system lies somewhere along the line joining the two particles, and its position depends on the relative masses of the particles and their distances from each other.

Physics Problem Solving

When faced with a physics problem, such as determining the correct statement about the center of mass in a two-particle system, it is crucial to follow a structured approach. This often involves evaluating provided statements, applying relevant formulas, and testing assumptions.

Firstly, review each statement against the governing equations. Consider the center of mass formula, which for any two masses, say \( m_1 \) and \( m_2 \), located at positions \( r_1 \) and \( r_2 \) respectively, is given by:

• \( R_{cm} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2} \)

Secondly, verify each statement against this formula. For example, a statement claiming equal distances applies only if masses are equal. Calculating each scenario allows for determining which scenarios are feasible.

Finally, compare mathematical results with conceptual understandings, ensuring consistency. Physics problem solving often needs a blend of calculation precision and theoretical comprehension, as in verifying the proportionalities or inversities stated in different options.

Proportionality in Physics

**Proportionality** is a key theme in physics and reappears in many forms across numerous topics, involving relationships between different quantities. In the context of the center of mass for a two-particle system, understanding how distances or other variables relate to masses is integral.

In this particular problem, one needs to consider how the distance from the center of mass to each particle relates to the mass of each particle. The correct statement from the problem demonstrates that this relationship is simple direct proportionality, meaning that the greater the mass of a particle, the closer the center of mass will be to it:

• If \( d_1 \) and \( d_2 \) are distances from the center of mass to particles, the formula \( \frac{R_{cm} - r_1}{r_2 - R_{cm}} = \frac{m_1}{m_2} \) holds true.

This equation showcases direct proportionality, simplifying how we anticipate the system's overall balance.

Remember, proportionality provides a systematic approach in physics to predict how changes in one aspect influence another, and understanding it equips us to solve complex real-world problems more intuitively.