Subtopic: Power transmission systems – gear trains, differential, final drives and power take-off
Question Statement:
A tractor gear box has 8 forward speeds. The speed ratios (number of engine revolutions for one revolution of driving wheel) vary in exact geometrical progression. If the speed ratios in highest and lowest gears are 14.9 and 108.8, respectively, the geometric constant is ___________.
Official Answer Key:
0.74 to 0.76
Step-by-Step Solution & Explanation
In gear-box terminology, the 'highest' gear (top/road gear, fastest travel speed) has the smallest speed ratio (14.9), and the 'lowest' gear (first/lowest travel speed) has the largest ratio (108.8). Numbering gears 1 (lowest, r₁=108.8) through 8 (highest, r₈=14.9) in geometric progression with common ratio K:
r₈ = r₁·K⁷ ⇒ K = (r₈/r₁)^(1/7) = (14.9/108.8)^(1/7) = (0.13695)^(1/7)
ln(0.13695) = −1.9880; /7 = −0.28400; K = e⁻⁰·²⁸⁴⁰⁰ = 0.753, within the accepted range.
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