Pathfinder Physics Solution — Laws of Motion (MCQ 35)

Corrected Technical Diagram (Perfect Geometry Alignment)

Method 1: Kinematic Constraint & Spring Elongation (Alternative Solution)

We can systematically solve this by establishing the internal relationship between individual spring extensions and the absolute rightward displacement x of the block.

1. Variable Definitions Based on System Topology

• Let the elongation of the top-left spring be x₁. Because this string is anchored directly to the wall, the center axle of the upper pulley shifts rightwards by exactly x₁.
• Let the elongation of the middle-right spring connected to the upper pulley axle be x₂.
• From the ideal force equilibrium conditions of the pulleys, the tensions dictate the remaining spring responses: the lower-left spring scales to an elongation of 2x₁, and the lower-right spring attached to the block scales to x₁.

2. Formulating the Constraint Geometry Equations

Tracking the net horizontal translation of the block through the length constraints of the upper and lower loop combinations gives us the boundary conditions:

Simplifying the compatibility condition directly isolates the ratio between the spring expansions:

3. Shifting Calculation of the Block

Expressing the total net shift x of the block explicitly using our base translation component x₁:

The external load $F$ dynamically pulls against a cumulative resisting hookean network equivalent to:

Substituting $x₁$ into our displacement relation delivers the geometric constraint solution:

Method 2: Work-Energy Equivalency

1. Total Network Potential Energy ($U$)

Summing the elastic strain energy stored concurrently across all four identical mechanical springs:

Substituting the load state parameters ($x₁ = \frac{F}{3k}$):

2. Work Done Integration

Because the external pulling force is introduced gradually starting from zero up to steady-state value $F$, the external energy work profile evaluates to:

Equating work input directly to total internal potential storage ($W = U$):