COM O23

Problem Statement: Identify the correct relation between the acceleration vectors $\vec{a}_A, \vec{a}_B, \vec{a}_C$ of the particles immediately after particle C is given a velocity $u$.

Analysis:

We consider the assembly of three rods and three particles as a single system. We examine the forces acting on this system immediately after the impulse has generated the velocity $u$.

1. External Forces:

• The impulse phase is over.
• Gravity is ignored.
• The pivot is described as “light” (massless) and free to move/rotate, implying no external constraints fixing it to the ground that would exert a net force after the impact.

Therefore, the net external force on the system is zero: $\Sigma \vec{F}_{ext} = \vec{0}$.

2. Newton’s Second Law for the System:

For a system of particles, the net external force equals the total mass times the acceleration of the Center of Mass ($\vec{a}_{CM}$).

$$ \Sigma \vec{F}_{ext} = M_{total} \vec{a}_{CM} = 0 \implies \vec{a}_{CM} = \vec{0} $$

3. Expanding the CM Acceleration:

The acceleration of the center of mass is the mass-weighted sum of individual accelerations:

$$ \vec{a}_{CM} = \frac{m_A \vec{a}_A + m_B \vec{a}_B + m_C \vec{a}_C}{m_A + m_B + m_C} $$

Since $\vec{a}_{CM} = \vec{0}$:

$$ m_A \vec{a}_A + m_B \vec{a}_B + m_C \vec{a}_C = \vec{0} $$

4. Substituting Masses:

Given $m_A = m, m_B = 2m, m_C = 3m$:

$$ m \vec{a}_A + 2m \vec{a}_B + 3m \vec{a}_C = \vec{0} $$ $$ \vec{a}_A + 2\vec{a}_B + 3\vec{a}_C = \vec{0} $$