Simple machine — wheel and axle (velocity ratio) For a simple wheel and axle, if D is the diameter of the effort wheel and d is the diameter of the load axle, what is the velocity ratio (V.R.)?

Introduction / Context:
The wheel and axle is a classic simple machine used to amplify an applied effort. Understanding its velocity ratio (V.R.) helps in computing mechanical advantage and efficiency, which are vital in lifting mechanisms and hoisting equipment.

Given Data / Assumptions:

• A rigid effort wheel of diameter D is keyed to a rigid axle of diameter d.
• There is no slip between rope and wheel/axle (pure rolling of the rope on the rim).
• Small motions are considered so linear travel equals the arc length at the rims.

Concept / Approach:
Velocity ratio is defined as V.R. = distance moved by effort / distance moved by load (for the same time interval). In one full revolution, the effort point on the wheel rim travels a circumference πD, while the load point on the axle rim travels πd. Hence V.R. = πD / πd = D / d.

Step-by-Step Solution:

Verification / Alternative check:
If D = d, V.R. = 1, meaning no amplification of motion; this matches intuition. If D is much larger than d, V.R. increases proportionally, indicating greater motion advantage.

Why Other Options Are Wrong:

• D + d and D − d: These do not arise from circumference ratios.
• D × d: Multiplies lengths, giving units of area, not a dimensionless ratio.
• d / D: This would be the inverse; it decreases as D grows, contradicting physical behavior.

Common Pitfalls:
Confusing mechanical advantage with velocity ratio. For an ideal machine, M.A. = V.R., but V.R. is defined purely by geometry and motion, independent of friction.

Final Answer:
D / d