Machine Design — Screw Jack Efficiency (Condition for Maximum) For a square-threaded screw jack, the efficiency becomes maximum at a specific helix angle α relative to the angle of friction φ. Choose the correct relation for α at maximum efficiency.

Introduction / Context: Screw jacks are classic simple machines used to lift loads through a threaded screw. Their efficiency depends on the helix angle α of the thread and the angle of friction φ. This question asks for the condition on α that maximizes efficiency for a square-threaded screw jack.

Given Data / Assumptions:

• Square-thread screw jack raising a load under steady conditions.
• Helix angle α is measured on the screw mean radius helix.
• Angle of friction φ is defined by tan φ = μ, where μ is the friction coefficient.
• Idealized geometry; no collar friction considered in the core relation.

Concept / Approach: For a screw jack, efficiency η during raising can be written in terms of α and φ. Maximizing η with respect to α gives a well-known design condition that balances mechanical advantage against frictional losses.

Step-by-Step Solution:

Verification / Alternative check: If φ = 0 (no friction), the condition yields α = 45°, which aligns with the symmetry of the tan-based expression and physical intuition that losses vanish at the geometric midpoint.

Why Other Options Are Wrong: '45° + φ/2' would increase α with friction, worsening efficiency. Expressions with '90° ± φ' do not arise from maximizing η for screw threads and are dimensionally unrelated to the optimum condition.

Common Pitfalls: Mixing definitions of α (lead vs helix angle), forgetting that the above result assumes square threads and neglects collar friction in the core optimization step.

Final Answer: α = 45° - φ/2.