Critical rotation speed of a ball mill: calculate the critical speed (revolutions per second) for a mill of diameter 1.2 m charged with 70 mm balls.

Introduction / Context:
The critical speed is the rotational speed at which the grinding media are centrifuged against the mill shell and cease to fall, eliminating grinding action. Mills are operated below this speed (typically 65–80% of critical).

Given Data / Assumptions:

• Mill diameter, D = 1.2 m → radius R = 0.6 m.
• Ball diameter = 70 mm → ball radius r_b = 0.035 m.
• Acceleration due to gravity g ≈ 9.81 m/s^2.

Concept / Approach:
Critical speed (in rps) considering the ball radius is n_c = (1 / 2π) * sqrt( g / (R − r_b) ). Using the effective radius gives a more accurate estimate than using R alone.

Step-by-Step Solution:
Compute effective radius: R_eff = R − r_b = 0.6 − 0.035 = 0.565 m.Compute inside square root: g / R_eff = 9.81 / 0.565 ≈ 17.37.Take square root: sqrt(17.37) ≈ 4.168 s^-1.Divide by 2π: n_c ≈ 4.168 / 6.283 ≈ 0.664 rps.

Verification / Alternative check:
Converting to rpm, n_c ≈ 0.664 * 60 ≈ 39.8 rpm, a typical critical speed for this size of mill.

Why Other Options Are Wrong:
0.50 or 1.00 rps: inconsistent with geometry and gravity.2.76 rps: far above critical for a 1.2 m mill.

Common Pitfalls:
Ignoring ball radius (using R only) slightly overestimates critical speed.

Final Answer:
0.66 rps