Locomotive balancing: the swaying couple (arising from out-of-phase unbalanced forces) attains its maximum or minimum value when the crank angle θ, measured from the line of stroke, equals which of the following?

Introduction / Context

Partial balancing of reciprocating masses in locomotives introduces lateral unbalanced forces on each side. Their difference creates a swaying couple that rocks the engine about a vertical axis. Knowing where this couple peaks helps assess track loading and ride quality.

Given Data / Assumptions

• Two-cylinder locomotive with cranks at right angles (a typical arrangement).
• Only a fraction of reciprocating mass is balanced on each side.
• Classical balancing theory (sinusoidal components) applies.

Concept / Approach

The horizontal unbalanced forces along the stroke for the two sides vary as cosθ and sinθ (phase-shifted by 90°). The swaying couple is proportional to their difference, giving a resultant term ∝ (cosθ − sinθ). The extrema of this function occur where its derivative vanishes, equivalently where sin(θ + 45°) = ±1.

Step-by-Step Solution

Verification / Alternative check

Using the identity sinθ − cosθ = √2·sin(θ − 45°) yields extrema at θ = 45° and 225° for the negative/positive phase choice; both formulations are consistent once the reference and sign conventions are fixed.

Why Other Options Are Wrong

• 90° and 180° / 180° and 270° / 270° and 360°: these correspond to zeros or intermediate values depending on phase choice; they are not the principal extrema for the standard swaying-couple form.
• 0° and 90°: not aligned with the extreme values of the cos–sin difference.

Common Pitfalls

• Confusing positions of maximum hammer blow (vertical dynamic load) with those of maximum swaying couple (lateral rocking).

Final Answer

45° and 225°