Velocity ratio — differential wheel and axle For a differential wheel and axle with effort wheel diameter D and axle diameters d1 (larger) and d2 (smaller), what is the velocity ratio (V.R.)?

Introduction / Context:
The differential wheel and axle increases motion advantage by using two axles of slightly different diameters wound in opposite senses. Understanding its velocity ratio is key to predicting lift per turn and selecting geometry.

Given Data / Assumptions:

• Effort wheel diameter D.
• Two axle diameters: d1 (larger), d2 (smaller), on the same shaft.
• Rope or chain winds on one axle and unwinds from the other; the load is supported such that rise equals half the net rope take-up.

Concept / Approach:
In one revolution, rope wound onto the larger axle = π d1; rope unwound from the smaller axle = π d2. The net rope shortening is π(d1 − d2). Since the load is supported by two parts of the rope, the load rises by half the net shortening.

Step-by-Step Solution:

Verification / Alternative check:
If d1 = d2, V.R. → ∞, which reflects no lift (mechanism locks), matching the physical expectation that there must be a diameter difference to produce motion.

Why Other Options Are Wrong:

• D / (d1 − d2): Misses the factor of 2 due to the two supporting rope segments.
• D / (d1 + d2), 2D / (d1 + d2): Use the sum instead of the difference; incorrect geometry.
• (d1 − d2) / (2D): Reciprocal of the correct expression.

Common Pitfalls:
Forgetting the “half” due to two supporting segments; confusing the differential wheel and axle with the plain wheel and axle or the Weston differential pulley.

Final Answer:
2D / (d1 − d2)