Problem 30 - Arihant AIEEE Physics

Question

Two small particles \(A\) and \(B\) of equal masses carrying equal positive charges are attached to the ends of a nonconducting light thread of length \(2 l .\) A particle \(C\) of mass twice of \(A\) is attached at mid-point of thread. The whole system is placed on a smooth horizontal floor and the particle \(C\) is given a velocity \(v\) as shown in the figure. Which of following statements is correct ? (a) The velocity of centre of mass of the system will remain constant during motion (b) At the instant of minimum separation between \(A\) and \(B\), there is no approach velocity between them or velocities of three particles are identical (c) The velocity of centre of mass of the system will be \(v / 2\) (d) All of the above

Step-by-Step Solution

Answer

All of the above (d) is correct.

Step 1: Calculate Mass of System

Let's begin by identifying the masses of our particles. Particles \(A\) and \(B\) have masses \(m\), and particle \(C\) has a mass of \(2m\). So, the total mass of the system is \(m + m + 2m = 4m\).

Step 2: Define Initial Conditions

Since the system is placed on a smooth horizontal floor, initially, only particle \(C\) is moving with velocity \(v\), and particles \(A\) and \(B\) are at rest. The initial velocity of the system's center of mass is due to particle \(C\).

Step 3: Calculate Initial Velocity of Center of Mass

The velocity of the center of mass \(v_{cm}\) can be calculated using the equation \(v_{cm} = \frac{m_A v_A + m_B v_B + m_C v_C}{m_A + m_B + m_C} = \frac{0 + 0 + 2m\cdot v}{4m}\), which simplifies to \(v/2\).

Step 4: Analyze System of Forces

There are no external forces acting on the center of mass horizontally, leading to the conservation of momentum of the system. Hence, the velocity of the center of mass will remain constant throughout the motion.

Step 5: Minimum Separation and Velocities

At the instant of minimum separation between \(A\) and \(B\), they are instantaneously at rest with respect to each other since they only have internal forces acting, which means there's no approach velocity between them, but not necessarily that all three velocities are identical.

Step 6: Evaluate Statements

Based on the calculations and analysis, statement (a) is correct as the velocity of the center of mass remains constant. Statement (b) is incorrect because even though there is no approach velocity between \(A\) and \(B\) at minimum separation, it does not imply that \(A\), \(B\), and \(C\) have the same velocity. Statement (c) is correct, the velocity of the center of mass is \(v/2\). Therefore, statement (d), "All of the above," is also correct.

Key Concepts

Center of Mass

In a system of particles, the center of mass is a crucial concept to understand. It is the point where the entire mass of a system could be concentrated. This point behaves as if all the external forces act directly upon it. In our example, we have particles \(A\), \(B\), and \(C\). Particle \(C\) moves initially, and its velocity influences the whole system's center of mass.To calculate the position of the center of mass (\( \vec{r}_{cm} \)), we use the formula: \[ \vec{r}_{cm} = \frac{m_A \vec{r}_A + m_B \vec{r}_B + m_C \vec{r}_C}{m_A + m_B + m_C} \]The velocity of the center of mass is similarly calculated by considering the momenta of the individual particles. Calculating the velocity of center of mass (\( v_{cm} \)) involves:

• Total mass of the system: \(4m\)
• Initial velocity contributions from \(C\): \(2mv\)

This leads to:\[ v_{cm} = \frac{2mv}{4m} = \frac{v}{2} \]This velocity remains constant if no external forces intervene.

Momentum Conservation

Momentum conservation is a fundamental principle in mechanics. It states that if no external forces are acting on a system, its total momentum remains constant. In the example provided, the entire system of particles lies on a smooth, frictionless surface, ensuring that no external horizontal forces alter the system's momentum.The conservation law can be expressed mathematically as:\[ \sum \vec{p}_{initial} = \sum \vec{p}_{final} \]Initially, only particle \(C\) has momentum as it is moving, while \(A\) and \(B\) are at rest. Therefore, the total initial momentum is:\[ \vec{p}_{initial} = 2m \cdot v \]As the system is closed and isolated in terms of horizontal movement, the total momentum remains:\[ \vec{p}_{total} = 2m \cdot v \]Consequently, despite internal forces between particles \(A\) and \(B\), the center of mass velocity doesn't change, staying consistent throughout the motion.

Particle Motion

Understanding particle motion within a system involves looking at how each part moves and interacts. For particles \(A\), \(B\), and \(C\), each has distinct characteristics that affect their motion.

• Particle \(A\) and \(B\) start at rest. Their motion is influenced by the internal forces within the thread system and potentially by particle \(C\).
• Particle \(C\) is initially given a velocity \(v\), leading the system's initial motion.

When analyzing such systems, consider internal and external forces. In this case, the absent horizontal external forces mean each particle's movements are dictated by the internal constraints and properties.At the instant of minimum separation between \(A\) and \(B\), these two particles lose their relative velocity towards each other since they only interact through internal forces, staying aligned on a tangential path with respect to their earlier motion.

System of Particles

A system of particles is a set of particles whose interactions and movements are considered collectively. Such systems are typically analyzed based on the distribution of mass, force interactions, and kinematic properties.In our example, the system includes particle \(C\), double the mass of \(A\) and \(B\), placed at the midpoint of a thread connecting \(A\) and \(B\).The uniform distribution of mass and the central placement of \(C\) leads to specific dynamic attributes:- The center of mass is calculated by considering the interim positions of all particles' masses.- Knowing the overall movement of the system, momentum distribution provides insight into how each part of the system might behave over time.When analyzing such systems, note that any motion imparted to the system, like the initial velocity to \(C\), primarily influences how the entire system moves initially. Over time, internal interactions cause subtle yet predictable behavior according to physics principles.