Critical speed Nc (rpm) of a ball mill, in terms of mill radius R1 and ball radius R2 (both in metres), is correctly given by which expression?

Introduction / Context:
Critical speed defines the rotational speed at which grinding media just cling to the mill shell. Above this speed, cataracting ceases and grinding action diminishes.

Given Data / Assumptions:

• R1 = mill radius, R2 = ball radius (m).
• g ≈ 9.81 m/s^2; unit conversion to rpm included.

Concept / Approach:
At the point of detachment, centripetal acceleration equals gravitational acceleration at the media center: ω^2 (R1 − R2) = g. Hence ω = sqrt(g/(R1 − R2)). Converting to rpm gives Nc = (60 / 2π) * sqrt(g/(R1 − R2)) ≈ 42.3 / sqrt(R1 − R2).

Step-by-Step Solution:
Set ω^2 (R1 − R2) = g.Solve ω = sqrt(g/(R1 − R2)).Convert to rpm: Nc = 60 ω / (2π) ≈ 42.3 / sqrt(R1 − R2).

Verification / Alternative check:
With D and d as diameters, the familiar form is Nc ≈ 42.3 / sqrt(D − d) (D,d in metres), consistent with the derived expression using radii.

Why Other Options Are Wrong:
Other constants or using (R1 + R2) are dimensionally/physically inconsistent.

Common Pitfalls:
Forgetting to subtract ball radius; the orbit center is at (R1 − R2).

Final Answer:
Nc = 42.3 / sqrt(R1 - R2)