Machine performance — relation between V.R. and M.A. when efficiency is constant If the efficiency of a lifting machine is kept constant, the velocity ratio (V.R.) is _____ proportional to the mechanical advantage (M.A.).

Introduction / Context:
Efficiency links force and motion characteristics of machines. Understanding how velocity ratio and mechanical advantage scale under constant efficiency is useful for design trade-offs.

Given Data / Assumptions:

• Efficiency η is constant for the comparisons.
• Mechanical Advantage M.A. = Load / Effort.
• Velocity Ratio V.R. = effort distance / load distance.

Concept / Approach:
The standard relation is η = M.A. / V.R. If η is constant, then M.A. is proportional to V.R., i.e., V.R. ∝ M.A. Therefore, V.R. is directly proportional to M.A. under fixed efficiency.

Step-by-Step Solution:

Verification / Alternative check:
Consider η = 0.5. If M.A. doubles, then V.R. doubles as well (V.R. = 2 * M.A.), confirming direct proportionality.

Why Other Options Are Wrong:

• Inversely proportional: Would require η ∝ M.A. * V.R., which is not the standard relation.
• Not / exponential / logarithmic: No basis in the machine efficiency equation.

Common Pitfalls:
Confusing the inverse relationship between M.A. and V.R. only in an ideal machine where η = 1 (in that special case M.A. = V.R., still a direct relation, not inverse).

Final Answer:
directly