Question

In a roving frame, the ratio of the diameter of the top (driver) cone-drum to the diameter of the bottom (driven) cone-drum is inversely proportional to bobbin diameter. At an instant,
Top cone-drum diameter = 212 mm
Bottom cone-drum diameter = 108 mm
Bobbin diameter = 54 mm
When the bobbin diameter becomes 100 mm, the ratio of the top cone-drum diameter to the bottom cone-drum diameter (up to 2 decimal places) is .................

Hint:

In roving frames, the cone-drum ratio reduces as bobbin diameter increases, ensuring proper winding tension control.

Answer 1

Step 1: Express the relationship.
The ratio is given as: \[ \frac{D_{top}}{D_{bottom}} \propto \frac{1}{D_{bobbin}} \] Step 2: Use initial conditions.
At bobbin diameter = 54 mm: \[ \frac{212}{108} = k . \frac{1}{54} \] \[ 1.96296 = \frac{k}{54} \] \[ k = 1.96296 \times 54 = 106.00 \; (\text{approx}) \] Step 3: Calculate for new bobbin diameter.
When bobbin diameter = 100 mm: \[ \frac{D_{top}}{D_{bottom}} = \frac{106}{100} = 1.06 \] Final Answer: The ratio lies between 1.04 and 1.08.