Machines — Velocity ratio of a Weston differential pulley block:\nIf D and d are the diameters of the larger and the smaller rigidly-connected pulleys, respectively, what is the velocity ratio (ideal conditions, no slip, no friction)?

Introduction / Context:
The Weston differential pulley (chain block) is a classic lifting machine. Its kinematics yield a characteristic velocity ratio (VR) that relates the distance moved by the effort chain to the distance moved by the load. Knowing VR is essential for estimating mechanical advantage and sizing equipment under ideal and real conditions.

Given Data / Assumptions:

• Larger pulley diameter = D; smaller pulley diameter = d.
• Pulleys are rigidly connected to rotate together (common angular speed).
• Idealized behavior: no slip, no friction, inextensible chain.
• Velocity ratio VR = distance moved by effort / distance moved by load.

Concept / Approach:
Because the two pulleys rotate together, the chain segments over the larger and smaller pitch circles move with different linear speeds. The net motion of the load depends on the differential take-up between these two circumferences. Under ideal conditions, the VR for a Weston differential pulley is expressed using the diameters (or radii) as a function of their difference.

Step-by-Step Solution:
Let circumferential speed on larger pulley = v_L; on smaller pulley = v_S.For the same angular speed ω: v_L = ω * (D/2) and v_S = ω * (d/2).Differential take-up per unit time = v_L − v_S = ω * (D − d)/2.The load is supported by two segments, so load speed = (v_L − v_S)/2 = ω * (D − d)/4.Effort chain speed corresponds to motion around the larger pulley (two strands engaged), effectively ω * D/2 on the hauling side; combining the standard derivation gives VR = (effort speed) / (load speed) = (ω * D/2) / (ω * (D − d)/4) = 2D / (D − d).

Verification / Alternative check:
Dimensionally, as D approaches d, (D − d) becomes small and VR grows very large, which matches the intuition that closer diameters give higher theoretical advantage but very slow load motion.

Why Other Options Are Wrong:

• D / (D − d): Misses the factor of 2 from two supporting strands.
• (D + d) / (D − d): Incorrect combination; does not follow from differential take-up.
• D / d: Belongs to simple belt speed ratios, not the Weston differential block VR.
• 2d / (D − d): Uses the smaller diameter in the numerator; not supported by derivation.

Common Pitfalls:

• Forgetting the two-strand support halves the load speed.
• Confusing torque/MA expressions with kinematic VR.

Final Answer:
2D / (D − d)