Neglecting the impact of fleet angle on the length of the rope, the minimum length, in m, of new winding rope is _____ (round off up to 2 decimals).

Steps to Solve

1. Identify the Relevant Measurements We know that the diameter of the drum and pulley is $1.2 , m$. This implies a radius of:

$$ r = \frac{1.2}{2} = 0.6 , m $$

The total vertical distance the rope travels is:

• From the bottom-most landing to the pulley: $215 , m$
• Height of the pulley (from the head gear structure): $32 , m$
• Therefore, the total height ($H$) is:

$$ H = 215 + 32 = 247 , m $$

2. Calculate the Length Along the Rope Next, we need to determine the length of rope based on the angle of $120^\circ$. The rope forms a triangle when it runs over the pulley. The length of the rope ($L$) in relation to the angle can be determined using the cosecant function.

To derive this mathematically, the relevant part of the rope is:

$$ L_{horizontal} = H + 0.5 , m $$ (adding half drum diameter for the horizontal length)

We find the sine of half the angle (since the triangle created is isosceles):

$$ \frac{L_{horizontal}}{\sin(60^\circ)} = L $$

3. Calculate the Final Length of the New Rope Substituting values into the formula gives:

$$ L = (H + 0.5) \cdot \frac{1}{\sin(60^\circ)} $$

Calculating this gives:

$$ L = (247.5) \cdot \frac{1}{\frac{\sqrt{3}}{2}} $$

4. Rounding Length Calculation After calculating $L$, round the result to two decimal places for the final answer.