Drilling machine speeds — geometric progression selection A six-speed drilling machine must cover drill diameters from 6.25 mm up to 25 mm at a cutting speed of 18 m/min. Assuming spindle speeds are arranged in a geometric progression to suit this range, what is the ratio between two consecutive spindle speeds (the common ratio)?

Introduction / Context:
Machine-tool speed charts are typically laid out in a geometric progression so that a single gearbox covers a wide range of cutting conditions without leaving big gaps. For drilling, spindle speed n is inversely proportional to drill diameter D for a given cutting speed V. This question tests selection of the common ratio for a six-speed drilling machine to span diameters from 6.25 mm to 25 mm at a fixed V.

Given Data / Assumptions:

• Diameter range: D_min = 6.25 mm, D_max = 25 mm.
• Cutting speed: V = 18 m/min (constant across the range).
• Number of distinct speeds: z = 6, arranged in geometric progression.
• Formula for drilling: n = V / (pi * D), with consistent units.

Concept / Approach:
Because n is proportional to 1/D, the ratio of extreme speeds equals the inverse ratio of extreme diameters. With z speeds in geometric progression, n_max = n_min * r^(z−1), where r is the common ratio. Solve for r from the required total range.

Step-by-Step Solution:
Compute the overall speed range: n_max / n_min = D_max / D_min = 25 / 6.25 = 4.Relate to geometric progression: r^(z−1) = r^5 = 4.Solve for r: r = 4^(1/5).Numerical value: 4^(1/5) ≈ 1.3195, which rounds to 1.32.

Verification / Alternative check:
If r = 1.32, then r^5 ≈ 1.32^5 ≈ 4.00, matching the required 4:1 speed range exactly, so the progression covers 6.25 to 25 mm properly.

Why Other Options Are Wrong:
1.02 gives an unrealistically fine progression (r^5 ≈ 1.10). 1.66 and 1.82 produce r^5 far exceeding 4, creating gaps. 1.41 gives r^5 ≈ 5.38, still too large a range.

Common Pitfalls:
Forgetting that speed varies as 1/D, using linear instead of geometric spacing, or mixing units (mm vs m) when computing absolute n. Here only ratios are needed, so unit consistency is implicit.

Final Answer:
1.32