1. Identifying Arrangement ‘S’

The system consists of a massless knot connecting three cords. For the knot to remain in equilibrium while the system accelerates, the transverse forces must cancel out. This requires a Symmetric Arrangement (labeled ‘S’).

In Arrangement S, the two cords connected to the masses are positioned symmetrically with respect to the line of the pulling force $\vec{F}$. Specifically, the geometry that satisfies the equilibrium condition results in each cord making a $30^\circ$ angle with the horizontal (line of action of $\vec{F}$).

2. Dynamics Equations

Equilibrium at the Knot:
Since the knot is massless, the sum of forces acting on it must be zero. Balancing the horizontal components: $$ \Sigma F_x = F – 2T \cos(30^\circ) = 0 $$ $$ F = 2T \left( \frac{\sqrt{3}}{2} \right) = T\sqrt{3} $$

Equation of Motion for the Mass:
Consider one of the masses $m$. The tension $T$ is the only force causing it to accelerate (ignoring vertical forces and friction): $$ T = ma $$

3. Solving for Acceleration

Substitute the expression for tension ($T = ma$) into the force equation: $$ F = (ma)\sqrt{3} $$ Solving for acceleration $a$: $$ a = \frac{F}{m\sqrt{3}} $$