Machines — Ideal machine relationship: In an ideal machine (no friction, no losses), the mechanical advantage (MA) is __________ the velocity ratio (VR).

Introduction / Context:Mechanical advantage (MA) and velocity ratio (VR) are cornerstone measures in machine theory. For ideal machines, energy is conserved perfectly, forging a simple relationship between input motion and output force that informs performance estimation and design trade-offs.

Given Data / Assumptions:

• Ideal machine: no friction, perfect rigidity, no elastic losses.
• MA = Load / Effort.
• VR = Distance moved by effort / Distance moved by load.
• Efficiency η = MA / VR.

Concept / Approach:Under ideal conditions, η = 1. Therefore MA = VR. Real machines have η < 1 due to friction and other losses, hence MA < VR in practice. The question explicitly specifies an ideal machine, so equality applies.

Step-by-Step Solution:Define efficiency: η = (Work output) / (Work input).For machines, η = MA / VR.Ideal machine ⇒ η = 1.Thus MA = VR.

Verification / Alternative check:Consider a simple lever or pulley block with ideal bearings and inextensible ropes. The kinematic ratio (VR) exactly predicts the achievable load amplification (MA), confirming equality when losses are absent.

Why Other Options Are Wrong:

• less than: True for real machines, not ideal.
• greater than: Violates energy conservation.
• undefined relative to: Relationship is well-defined via efficiency.

Common Pitfalls:

• Using real-world intuition (friction) for an idealized theoretical question.
• Mixing up definitions of MA and VR.

Final Answer:equal to