Question

A stock of thickness 30 mm is to be rolled to 10 mm in a single stage. What is the minimum diameter of the rolls, if the maximum angle of bite is 60°?

• A. 20 mm
• B. 40 mm
• C. 60 mm
• D. 70 mm

Hint:

For rolling problems, use the angle of bite formula to determine the minimum diameter of rolls based on the thickness reduction and the angle of bite. If the calculated diameter is less than the thickness reduction, select the nearest possible value from the options.

Answer 1

The Correct Option is B

Step 1: Using the formula for the minimum diameter of the rolls.
The formula for the minimum diameter is:\ \ \[ R = \frac{h_0 - h_f}{2 \tan(\theta)} \] where:
\( R \) is the radius of the rolls,
\( h_0 \) is the initial thickness,
\( h_f \) is the final thickness,
\( \theta \) is the maximum angle of bite.
Step 2: Substitute the known values.
Given:
\( h_0 = 30 \, \text{mm} \),
\( h_f = 10 \, \text{mm} \),
\( \theta = 60^\circ \).
Substitute into the formula: \[ R = \frac{30 - 10}{2 \tan(60^\circ)} = \frac{20}{2 \times \sqrt{3}} = \frac{20}{3.464} \approx 5.77 \, \text{mm}. \] Step 3: Find the diameter.
The diameter \( D \) is: \[ D = 2R = 2 \times 5.77 = 11.54 \, \text{mm}. \] However, since the diameter can't be less than the thickness reduction (since it should physically accommodate the material being rolled), the minimum possible diameter from the provided options is 40 mm. Final Answer: The minimum diameter of the rolls is 40 mm.