The problem asks to find the ratio of beat-up force between two shuttle looms, Loom A and Loom B, given their respective crank radii ($r$) and eccentricity ratios ($e$). The looms operate at the same picks per minute and have identical sley masses and associated systems.

Shuttle Loom Beat Up Force Ratio

Identifying Key Parameters

• Loom A: Crank radius $r_A = 10 \text{ cm}$, Eccentricity ratio $e_A = 0.5$
• Loom B: Crank radius $r_B = 6 \text{ cm}$, Eccentricity ratio $e_B = 0.4$
• Assumptions: Same picks per minute, same sley mass and system for both looms.

Determining Beat Up Force Relationship

For shuttle looms, the beat-up force ($F$) is influenced by several factors including the mass of the sley, the speed of operation, the crank radius ($r$), and the mechanism's geometry, often characterized by the eccentricity ratio ($e$). A common approximation relating these factors for the beat-up force is proportional to the product of the crank radius and a term involving the eccentricity ratio:

$F \propto r(1+e)$

Since the mass and operating speed are the same for both looms, the ratio of their beat-up forces will depend only on the variation in $r$ and $e$.

Calculating the Ratio of Forces

The ratio of the beat-up force of Loom A to Loom B ($F_A / F_B$) can be expressed as:

$ \frac{F_A}{F_B} = \frac{r_A(1+e_A)}{r_B(1+e_B)} $

Substitute the given values:

$ \frac{F_A}{F_B} = \frac{10 \text{ cm} \times (1 + 0.5)}{6 \text{ cm} \times (1 + 0.4)} $

Simplify the expression:

$ \frac{F_A}{F_B} = \frac{10 \times 1.5}{6 \times 1.4} $

$ \frac{F_A}{F_B} = \frac{15}{8.4} $

Final Beat Up Force Ratio

Perform the division:

$ \frac{15}{8.4} \approx 1.7857 $

Rounding the result to 1 decimal place gives 1.8.

This value (1.8) lies within the provided range of 1.7 to 1.9.