In basic machines (Engineering Mechanics): What is the velocity ratio (VR) of a differential wheel and axle with wheel radius R and two axle radii r1 and r2 (r1 ≠ r2)?

Introduction / Context:
The differential wheel and axle is a classic lifting machine studied in Engineering Mechanics and Strength of Materials. It combines two axles of slightly different radii on the same spindle with a common wheel. Understanding its velocity ratio (VR) is essential for predicting ideal mechanical advantage (IMA) and for comparing theoretical performance versus real efficiency with friction.

Given Data / Assumptions:

• Wheel radius = R.
• Two axle radii = r1 and r2, with r1 ≠ r2.
• Light, inextensible rope; idealized, frictionless pulleys for the VR derivation.
• Small differential between r1 and r2 to obtain high VR.

Concept / Approach:
In one revolution, the rope wound on the larger axle shortens the load side, while the rope unwound from the smaller axle lengthens it, producing a net downward movement of the load equal to the difference of the circumferences on the two axles. Input movement equals the circumference at the wheel. VR is the ratio of input displacement to load displacement in one revolution.

Step-by-Step Solution:

Verification / Alternative check:
If r1 approaches r2, the denominator becomes small and VR becomes large, which matches the practical purpose of using a differential arrangement for higher mechanical advantage.

Why Other Options Are Wrong:

• B and D: Use a sum r1 + r2, which does not reflect the differential action.
• C: Inverts the correct relationship and places 2R in the denominator.
• E: Misses the factor 2 from the two rope segments supporting the load.

Common Pitfalls:
Forgetting that two rope segments support the load and confusing circumference differences with radius differences. Real systems include friction; thus actual mechanical advantage < VR.

Final Answer:
VR = 2R / (r1 - r2)